tag:theconversation.com,2011:/africa/topics/puzzles-2284/articlesPuzzles – The Conversation2023-04-12T11:21:12Ztag:theconversation.com,2011:article/2030822023-04-12T11:21:12Z2023-04-12T11:21:12ZTetris movie: why the story of the game’s origins is legendary<figure><img src="https://images.theconversation.com/files/519796/original/file-20230406-217-4qt847.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C1366%2C768&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">There have been numerous iterations of Tetris since the game was first introduced but the iconic shapes never change. </span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/110457687@N03/15538236401">Downloadsource.es/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span></figcaption></figure><p>On my bedside cabinet, next to my alarm clock, is a jar holding my cufflink collection. One set contains seven odd cufflinks. They are bold in colour, now a bit scratched with flaking paint, but with clear geometric designs: a squat Z, S and T, an L, a J, a square and lastly, the ever useful long bar.</p>
<p>Even as a lecturer in games development, I don’t tend to wear my game affiliations that boldly but these cufflinks, the odd badge and my Minecraft waistcoat are exceptions. There are very few video game elements that I could describe so simply that even some non-gamers would recognise. But I am, of course, talking about the shapes, or the “tetrominoes”, from the nearly 40-year-old game of Tetris. </p>
<p>Alongside Pac-Man, Super Mario, and Sonic the Hedgehog, Tetris was one of the first video games to break into popular culture. How else can you explain the recent release of the <a href="https://www.apple.com/tv-pr/news/2023/02/apple-original-films-unveils-trailer-for-tetris-new-thriller-starring-taron-egerton/">Tetris movie</a>, starring Taron Egerton? Bizarrely, this film is based on what you think would be the rather dry legal arguments of the intellectual property rights of the game.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/-BLM1naCfME?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The official trailer for the Tetris movie on Apple TV+</span></figcaption>
</figure>
<p>But within games lore, the tale of how Tetris came to be is legendary. </p>
<p>Alexey Pajitnov, a speech recognition researcher at the Soviet Union’s Academy of Sciences, developed a range of puzzle games in the early 1980s, by “borrowing” spare time on his workplace’s Electronika 60 computer. </p>
<p>The machine had no graphical display and so the games were displayed using text but Tetris still hooked many of Pajitnov’s colleagues. Soon it was on most computers in various Soviet organisations.</p>
<p>Pajitnov wanted to share his game, but this being the late Soviet Union era, he had little idea of how game publishing worked and his employers weren’t pleased about the “wasted” time on their expensive computer. Plus Soviet copyright law gave the state control over the software. </p>
<p>However, Pajitnov negotiated the rights to the Academy via his supervisor, who sent the game to Hungarian game publisher Novotrade. That led to Tetris seeing limited success behind the iron curtain.</p>
<figure class="align-left ">
<img alt="A bearded man wearing a white top and jeans stands smiling on a blue stage with his hands raised." src="https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/519618/original/file-20230405-18-wj9exs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Tetris creator, Alexey Pajitnov.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/imaginecup/9271942618/in/photostream/">ImagineCup/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>In Hungary, Robert Stein of Andromeda Software saw Tetris, liked it and approached Pajitnov about obtaining the rights. Pajitnov responded via fax that he was interested. Stein took that fax and without drawing up a contract, proceeded to sell the rights at the 1987 Las Vegas Consumer Electronics Show. </p>
<p>Tetris became a massive success, being ported to multiple platforms and winning multiple awards. But what then followed was a protracted legal battle which stretched across continents, involved several gaming companies and numerous iterations of Tetris itself. There were versions on the <a href="https://www.computinghistory.org.uk/det/1227/Amstrad-Plc/">Amstrad</a>, the <a href="https://www.computinghistory.org.uk/det/182/Acorn-BBC-Micro-Model-B/">BBC Micro</a> and the <a href="https://americanhistory.si.edu/collections/search/object/nmah_334638">Apple II</a> to name but three. </p>
<p>Eventually, in the late 1980s, Nintendo showed an interest in wanting to obtain Tetris for their upcoming <a href="https://www.nintendo.co.uk/Hardware/Nintendo-History/Game-Boy/Game-Boy-627031.html">Game Boy</a> console. Since then, it’s practically been ubiquitous. </p>
<figure class="align-center ">
<img alt="A grey square Tetris cartridge propped up against a Nintendo Game Boy console" src="https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&rect=53%2C8%2C6000%2C3979&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/519611/original/file-20230405-16-nn8pa9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Nintendo’s Game Boy was released in 1989 and Tetris became the most popular game.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Game_Boy_and_Tetris.jpg">Sammlung der Medien und Wissenschaft/Wikimedia</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Quite simply, Tetris is one of the most engaging computer games ever devised. Some have tried to pin <a href="https://www.semanticscholar.org/paper/Trauma%2C-treatment-and-Tetris%3A-video-gaming-volume-Butler-Herr/2382b9b18e1a9fafb228f94c3ab1b7504b5b3e7d">psychological aspects</a> to it. The slow ramping up of difficulty is not just due to the increasing speed but also because your failures stay to thwart you and victories are therefore fleeting. </p>
<p>For those of us old enough to have grown up with the first generation of home computers such as the <a href="https://worldofspectrum.org">ZX Spectrum</a>, Tetris is our “when I were a lad” type of game. I still remember the drama of the thumping and ominous beat of the music on the Commodore 64, forgoing the catchy Russian folk song of the original.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/Rpt-G3dRcek?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">What Tetris looked and sounded like on the Commodore 64.</span></figcaption>
</figure>
<p>It was a time when you couldn’t rely on the Oscar winning performances of your voice actors, the ultra-realism of your graphics or a sumptuous orchestral soundtrack. It was beeps and limited colours of obvious pixels, so the gameplay had to grab you even more to get that “just one more go” kick. </p>
<p><a href="https://doom.fandom.com/wiki/Doom">Doom</a> is often cited as being the game featured across all platforms but I’d argue that Tetris really holds that crown. Having first played it on my <a href="https://www.commodore.ca/commodore-products/commodore-64-the-best-selling-computer-in-history/">Commodore 64</a>, I went on to play it on my <a href="http://theamigamuseum.com/amiga-models/amiga-1000/">Amiga</a>, many iterations on PC, then the Xbox 360. It’s available on the current generation of consoles, on phones and there is even a virtual reality version. All with the same essential gameplay. You can’t really mess with near perfection.</p>
<p>Having grown up with <a href="https://www.onrec.com/news/news-archive/what-is-8-bit-graphics-and-how-it%E2%80%99s-used-nowadays">8-bit graphics</a>, it’s fascinating to watch my students, born decades after Tetris, copying that retro style for their modern games design.</p>
<figure class="align-center ">
<img alt="A computer game animation shows a platform set against a blue sky, where various coloured blocks are mounted up." src="https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/519808/original/file-20230406-24-7ntf7e.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The 2012 game, Fez, was designed by Phil Fish.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Fez_%28video_game%29_screenshot_05.png">Polytron Corporation</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>When Phil Fish designed his game <a href="http://fezgame.com">Fez</a>, a successor to Tetris as a video game puzzler, he wanted to depict the remains of an ancient civilisation, but a video game ancient civilisation. </p>
<p>Look closely at the stone ruins of this long-dead race. They are made from recognisable, odd-shaped blocks: a squat Z, S and T, an L, a J, a square and lastly the ever useful long bar.</p><img src="https://counter.theconversation.com/content/203082/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Simon Scarle does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Cold war intrigue and an international legal fight were behind one of the most popular video games ever.Simon Scarle, Senior Lecturer in Games Development, Cardiff Metropolitan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1905292022-12-28T21:09:29Z2022-12-28T21:09:29ZThe history and mystery of Tangram, the children’s puzzle game that harbours a mathematical paradox or two<figure><img src="https://images.theconversation.com/files/498377/original/file-20221201-20-cw8scj.jpg?ixlib=rb-1.1.0&rect=0%2C39%2C6630%2C4337&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Have you played the puzzle game Tangram? </p>
<p>I remember, as a child, being fascinated by how just seven simple wooden triangles and other shapes could offer endless entertainment. Unlike LEGO, the Tangram pieces do not snap together, and unlike the pieces of a jigsaw puzzle, they do not form a painted picture.</p>
<p>Instead, Tangram invites you to fit all the pieces together to form countless varieties of shapes. You can make your own shapes or you can try to form shapes that others have created. For instance, here’s one way to form a swan shape using Tangram pieces:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A swan shape in Tangram." src="https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">This is one of several ways to make a swan shape using Tangram. Can you find another?</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>But it’s not the only way to make a swan. Can you find others? If you do not have the physical puzzle at hand, you can <a href="https://toytheater.com/tangram/">use</a> a virtual version of Tangram.</p>
<p>Tangram is accessible and yet challenging, and an excellent <a href="https://link.springer.com/article/10.1007/BF02354839">educational tool</a>. It’s still <a href="https://education.nsw.gov.au/teaching-and-learning/curriculum/mathematics/mathematics-curriculum-resources-k-12/mathematics-k-6-resources/how-to-make-a-tangram">used</a> in <a href="https://education.nsw.gov.au/teaching-and-learning/curriculum/mathematics/mathematics-curriculum-resources-k-12/mathematics-k-6-resources/how-to-make-a-tangram">schools</a> today to help illustrate mathematical concepts and develop mathematical thinking skills. It even features a paradox or two.</p>
<hr>
<p>
<em>
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Read more:
<a href="https://theconversation.com/5-math-skills-your-child-needs-to-get-ready-for-kindergarten-103194">5 math skills your child needs to get ready for kindergarten</a>
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</em>
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<hr>
<h2>A long history of rearrangement puzzles</h2>
<p>Tangram is one of many rearrangement puzzles that have appeared throughout the ages. The earliest known rearrangement puzzle, the <a href="https://mathworld.wolfram.com/Stomachion.html">Stomachion</a>, was invented by Greek mathematician Archimedes 2,200 years ago and was popular for centuries among Greeks and Romans. </p>
<p>It consists of 14 puzzle pieces that can fit together in the form of many different shapes. There are <a href="https://mathweb.ucsd.edu/%7Efan/stomach/tour/stomach.html">536 different ways</a> to fit the pieces together as a square. </p>
<p>Then there’s the <a href="https://www.mathpuzzle.com/eternity.html">Eternity Puzzle</a>, released in 1999, which consists of 209 blue puzzle pieces that together form a big circle-like shape. It was very popular and sold <a href="https://en.wikipedia.org/wiki/Eternity_puzzle">500,000 copies</a> worldwide, perhaps due to the 1 million British pounds promised to whoever first solved it. </p>
<p>Less than a year later, the mathematicians Alex Selby and Oliver Riordan <a href="https://plus.maths.org/content/prize-specimens">solved the puzzle</a> and claimed the prize. The creator of the puzzle, the <a href="https://en.wikipedia.org/wiki/Christopher_Monckton,_3rd_Viscount_Monckton_of_Brenchley">controversial</a> Christopher Monckton, said at the time he had to <a href="http://news.bbc.co.uk/1/hi/entertainment/992393.stm">sell his house</a> to raise the prize money. </p>
<p>The origins of Tangram stretch back to the third century Chinese mathematician <a href="https://en.wikipedia.org/wiki/Liu_Hui">Liu Hui</a>. Among many other <a href="https://www.jstor.org/stable/2691200">accomplishments</a>, Liu Hui used rearrangements of geometrical shapes to elegantly explain mathematical facts such as the <a href="https://en.wikipedia.org/wiki/Pythagorean_theorem">Gougu Rule</a>, also known as Pythagoras’ Theorem. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Rearrangement proof of Pythagorean theorem" src="https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=345&fit=crop&dpr=1 600w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=345&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=345&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=433&fit=crop&dpr=1 754w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=433&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=433&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Shapes can be rearranged to explain the Gougu Rule, also known as Pythagoras’ Theorem.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Animated_gif_version_of_SVG_of_rearrangement_proof_of_Pythagorean_theorem.gif">Animation by William B. Faulk, CC BY-SA 4.0, via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>This rearrangement approach to geometry was later evident in the creation of 12th century <a href="https://www.wired.com/2011/01/games-for-the-hands-and-mind-chinese-puzzles-at-the-moca/">Chinese banquet tables</a> (rectangular tables designed to be arranged into patterns that might please or entertain dinner guests).</p>
<p>A different version, known as a <a href="https://www.logicagiochi.com/en/the-history/">butterfly table</a>, was popularised in the early 17th century and featured a broader variety of shapes. A surviving table set can be seen in the <a href="https://www.chinadiscovery.com/jiangsu/suzhou/lingering-garden.html">Lingering Garden (Liu Yuan)</a> which is part of a <a href="https://whc.unesco.org/en/list/813/">UNESCO World Cultural Heritage</a> site in Suzhou.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A Tangram puzzle lies on a table." src="https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The Tangram was popularised as a puzzle game around the year 1800.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<h2>The Tangram craze</h2>
<p>According to <a href="https://www.amazon.com/Tangram-Book-Jerry-Slocum/dp/1402704135">The Tangram Book</a> by Jerry Slocum and other authors, the Tangram was popularised as a puzzle game around the year 1800. </p>
<p>They report the inventor, an unknown Chinese person using the pen name Yang-Cho-Chu-Shih (“Dimwitted recluse”), published Ch'i chi'iao t'u (“Pictures Using Seven Clever Pieces”), a book containing hundreds of Tangram puzzle shapes. </p>
<figure class="align-right ">
<img alt="Patterns from a Tangram puzzle and solution books, China c. 1815 (British Library 15257.d.5, 15257.d.14)" src="https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=968&fit=crop&dpr=1 600w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=968&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=968&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1216&fit=crop&dpr=1 754w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1216&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1216&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Patterns from a Tangram puzzle and solution books, China c. 1815 (British Library 15257.d.5, 15257.d.14)</span>
<span class="attribution"><span class="source">British Library</span></span>
</figcaption>
</figure>
<p>This sparked a craze for the game in China. Other Tangram puzzle books were soon published, with some eventually making their way to Japan, the United States and England, where they were translated and extended. </p>
<p>During 1817-18, the Tangram <a href="https://collections.libraries.indiana.edu/lilly/exhibitions_legacy/collections/overview/puzzle_docs/Tangram-Worlds_First_Puzz_Craze.pdf">craze</a> spread like <a href="https://www.puzzlemuseum.com/month/picm09/2009-03-early-tangram.htm">wildfire</a> to France, Denmark and other European countries. Worldwide interest in Tangram has endured ever since. </p>
<h2>An educational tool harbouring a paradox or two</h2>
<p>The lasting popularity of Tangram might partly be due to it allowing so many shapes with so few pieces. </p>
<p>Researchers have found that Tangram can help students’ <a href="https://journaljesbs.com/index.php/JESBS/article/view/765">visual and geometric thinking</a> and even their <a href="https://www.tandfonline.com/doi/abs/10.1080/15248372.2012.725186">arithmetic skills</a>.</p>
<p>Tangram may help in the assessment of children’s learning of <a href="https://journals.sagepub.com/doi/10.1177/2332858419829723">written languages</a> and of their <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6798005/">emotional regulation skills</a>.</p>
<p>For most people, though, Tangram is just a fun and creative challenge.</p>
<p>There are also some Tangram “paradox” puzzles discussed in <a href="https://www.amazon.com/Tangram-Book-Jerry-Slocum/dp/1402704135">The Tangram Book</a> and elsewhere online, where Tangram pieces are arranged to make two seeming identical shapes (but where one appears to have a leftover piece). </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="The Monk puzzle" src="https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=505&fit=crop&dpr=1 600w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=505&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=505&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=634&fit=crop&dpr=1 754w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=634&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=634&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The two monks Tangram paradox.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Two_monks_tangram_paradox.svg">AlphaZeta, CC0, via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Can you explain the “paradox” – why one has a triangular “foot” and the other does not, even though both images use all seven pieces? </p>
<p>As a bonus challenge, perhaps you can you solve the similar infinite chocolate bar “paradox” popularised on Instagram and TikTok.</p>
<p><iframe id="tc-infographic-795" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/795/af484f026421a1a75b5436ba26c883774684659d/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>Good luck and happy puzzling!</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/learn-how-to-make-a-sonobe-unit-in-origami-and-unlock-a-world-of-mathematical-wonder-171390">Learn how to make a sonobe unit in origami – and unlock a world of mathematical wonder</a>
</strong>
</em>
</p>
<hr>
<img src="https://counter.theconversation.com/content/190529/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The Conversation bought the author a Tangram set to play with so he could write this article.</span></em></p>Tangram is accessible yet challenging, and an excellent educational tool. It’s still used in schools today to help illustrate mathematical concepts and develop mathematical thinking skills.Thomas Britz, Senior Lecturer, UNSW SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1763252022-02-04T13:09:21Z2022-02-04T13:09:21ZWant to master Wordle? Here’s the best strategy for your first guess<figure><img src="https://images.theconversation.com/files/444360/original/file-20220203-17-18vqztc.jpg?ixlib=rb-1.1.0&rect=0%2C5%2C3514%2C2334&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">There are 2,315 five-letter words in Wordle's dictionary.</span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/news-photo/wordle-game-displayed-on-a-phone-and-a-laptop-screens-is-news-photo/1237931947?adppopup=true">Jakub Porzycki/NurPhoto via Getty Images</a></span></figcaption></figure><p>As Wordle has <a href="https://www.nytimes.com/2022/01/31/crosswords/nyt-wordle-purchase.html">skyrocketed in popularity</a>, multiple media outlets <a href="https://slate.com/technology/2022/01/wordle-how-to-win-strategy-crossword-experts.html">have published articles</a> <a href="https://www.wsj.com/articles/wordle-has-tur">that explore the best word</a> to use as your initial guess. </p>
<p>Often the authors of these pieces theorize that the word ought to be one that uses as many vowels as possible, contains letters that frequently appear in English or possesses features that regularly occur in the language.</p>
<p>Well, my finance students and I decided to tackle this question in as definitive a manner as possible by determining the optimal first word to play in Wordle. </p>
<p>Our analysis actually ran through all possible combinations of five-letter words and ran simulations across all possible iterations – over 1 million of them – to figure out the best starting strategy.</p>
<h2>A ‘tried’ and true approach</h2>
<p><a href="https://www.cnet.com/how-to/wordle-explained-what-you-need-to-know-about-the-viral-word-game/">In Wordle</a>, players have six attempts to guess a five-letter word. Each time the player makes a guess, they learn whether each letter is correct and in the right location, appears in the word in another location or isn’t in the word at all.</p>
<p>Players can have different approaches. Some might simply want to solve the word, even if it takes six tries. Others try to do it in as a few guesses as possible.</p>
<p>Based on our analysis, if you’re trying to win in as few guesses as possible, the top three words to go with are “slice,” “tried” and “crane.” Using any of these three words will produce an average number of word attempts of 3.90, 3.92, and 3.92, respectively, if you’re using an optimal strategy to play (more on that later).</p>
<p><iframe id="LTbA9" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/LTbA9/4/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>If, on the other hand, you’re simply trying to win within the allotted six guesses, the top three words to play are “adept,” “clamp” and “plaid.” Using any of these three words will yield an average success rate in winning the game of 98.79%, 98.75%, and 98.75%, respectively, if you’re playing the optimal strategy.</p>
<p><iframe id="MuBec" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/MuBec/4/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>And herein lies the first interesting distinction between playing to win and playing to win in as few guesses as possible. </p>
<p>If you’re playing to win in the allotted six guesses, it appears best to play a word that has just one vowel and four consonants in it, as six out of the top 10 words have just one vowel. But if you’re playing to win in as few guesses as possible, it’s best to play a word that has two vowels and three consonants: All of the top 10 have two vowels.</p>
<h2>Inside the simulations</h2>
<p>Other researchers, <a href="https://theconversation.com/wordle-the-best-word-to-start-the-game-according-to-a-language-researcher-175114">such as David Sidhu at University College London</a>, have tried to determine the “best first word” from a linguistic perspective. In these efforts, the best selection is decided by how often certain letters appear in the English language, or the frequency of where these letters are located in five-letter words. </p>
<p>While these approaches are noble, our analysis extends beyond them by actually performing simulations across all possible word options to find the best type of word to play first.</p>
<p>To perform this analysis, two of my students, Tao Wei and Kanwal Ahmad, constructed a program that went through all <a href="https://slate.com/technology/2022/01/wordle-how-to-win-strategy-crossword-experts.html">2,315 official five-letter words</a> in Wordle’s dictionary. The program attempted each possible word as a first guess and ran simulations across all possible end word solutions, checking how long each attempt would take to guess the correct end word – 1,692,265 total simulations. </p>
<p>We then averaged all attempts for each word to see how many guesses one could expect to make to get to the correct end word. </p>
<p>To perform this massive simulation requires a method for picking the optimal word on the second guess, third guess and so on. </p>
<p>To give yourself the best odds on each ensuing guess, it’s important to select letters that are most likely to appear in each position. So the program used the list of 2,315 total words to determine the frequency at which each letter appears. </p>
<p>After receiving the results from the previous guess, the program filtered down the possible words to those that meet the criteria. Say the first guess were “bloke,” and L and E were in the correct position, while B, O and K didn’t appear in the solution. The program would then narrow down the list of possible words to those like “flume” and “slate.”</p>
<p>The program then assigns a score to each word in this list, where the score is the sum of the frequency of its letters. The word “slate,” for example, has a score of 37% because the letter “S” appears 5% of the time in the full list, while the letter appears “A” 8% of the time, and so on. The word with the highest score is then submitted as the next guess.</p>
<p>Running this simulation over all possible first guesses and against all possible solutions yielded the results.</p>
<p>But maybe you don’t want to start with the same word every time you play. In that case – and if you want to win with the fewest guesses – try making sure your first guess has two vowels, with one of them at the end of the word.</p>
<p>If you’re just looking to win within the allotted six guesses, then you may want to consider a word with fewer vowels – and definitely a word that ends in a consonant.</p>
<p>Hopefully our mathematical approach to Wordle hasn’t sucked all the joy out of the game. At the very least, it’ll give you a leg up if you decide to put a friendly wager on tomorrow’s game.</p>
<p>[<em>Get fascinating science, health and technology news.</em> <a href="https://memberservices.theconversation.com/newsletters/?nl=science&source=inline-science-fascinating">Sign up for The Conversation’s weekly science newsletter</a>.]</p><img src="https://counter.theconversation.com/content/176325/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Derek Horstmeyer does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Whether you want to win with as few guesses as possible, or you just want to figure out the right word before running out of turns, a scholar offers some tips.Derek Horstmeyer, Professor of Finance, George Mason UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1752272022-01-25T15:46:31Z2022-01-25T15:46:31ZThe Wordle craze: Why do we love puzzles, and are they good for our brains?<figure><img src="https://images.theconversation.com/files/442097/original/file-20220123-27-14cfsw6.jpg?ixlib=rb-1.1.0&rect=44%2C0%2C5000%2C3323&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Wordle is the latest word game to captivate millions. </span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><iframe style="width: 100%; height: 175px; border: none; position: relative; z-index: 1;" allowtransparency="" src="https://narrations.ad-auris.com/widget/the-conversation-canada/the-wordle-craze--why-do-we-love-puzzles--and-are-they-good-for-our-brains" width="100%" height="400"></iframe>
<p>In recent weeks, a web-based word puzzle called <a href="https://www.powerlanguage.co.uk/wordle/">Wordle</a> has become a popular daily distraction. Suddenly, <a href="https://www.nytimes.com/2022/01/03/technology/wordle-word-game-creator.html">millions of people are focused on their vocabulary of five-letter words</a>, and are newly aware of concepts like letter frequency and letter position as they strategize about the best opening words and faster solutions. </p>
<p>For these people, Wordle is captivating. Previous research can help us understand how our brains respond to word games, and why we love them.</p>
<p>Wordle is a single-player puzzle that combines elements of several games, including Scrabble and Battleship. My colleagues and I <a href="https://globalnews.ca/news/145744/u-of-c-researchers-investigate-what-makes-accomplished-scrabble-players-so-s-m-a-r-t/">have studied Scrabble as a way of understanding how language is processed in the brain</a>, and how that processing changes with experience.</p>
<h2>This is your brain on Scrabble</h2>
<p>Competitive Scrabble players are people who spend a great deal of time playing Scrabble, competing in Scrabble tournaments, memorizing word lists and practising anagramming — shuffling sets of letters to create different words. </p>
<p>Much like chess players, competitive Scrabble players are <a href="https://www.wespa.org/aardvark/cgi-bin/rating.cgi">ranked in an international rating system</a> based on tournament results. We recruited competitive players from Scrabble tournaments and clubs and gave them a series of tasks to understand how all of this Scrabble practice and play alters their mental processes.</p>
<p>In our first study, we found that <a href="https://doi.org/10.3758/s13421-011-0137-5">competitive Scrabble players recognized words faster than those who didn’t routinely play Scrabble, particularly when words were presented vertically</a>. Vertical word presentation is unusual in written English but common in Scrabble, and competitive players are very good at recognizing vertical words. </p>
<p>We also found that Scrabble players quickly recognized words without fully processing word meaning. This is probably because in Scrabble, you need to know whether different strings of letters make up legal plays, but you don’t actually need to know what those words mean.</p>
<figure class="align-center ">
<img alt="A Scrabble player places tiles to spell a word on a green Scrabble board." src="https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=382&fit=crop&dpr=1 600w, https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=382&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=382&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=479&fit=crop&dpr=1 754w, https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=479&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/442098/original/file-20220123-23-ugabse.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=479&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A Scabble player places tiles to spell a word during a meeting of the Vancouver Scrabble Club.</span>
<span class="attribution"><span class="source">THE CANADIAN PRESS/Darryl Dyck</span></span>
</figcaption>
</figure>
<p>We also used brain imaging to study how <a href="https://doi.org/10.1016/j.cortex.2015.03.015">all those years of intensive practice might have altered brain processes for language in competitive Scrabble players</a>. </p>
<p>We found that when recognizing words and making simple decisions about them, competitive Scrabble players used a different network of brain areas than those who didn’t play Scrabble competitively. Scrabble experts made use of brain regions not typically associated with word meaning retrieval, but rather those associated with visual memory and perception.</p>
<h2>A Scrabble habit makes you … good at Scrabble</h2>
<p>We wondered whether the effects of Scrabble practice that we observed in competitive players have benefits beyond Scrabble. Does playing lots of Scrabble make you good at anything else? The answer seems to be no. </p>
<p>We investigated that question by <a href="https://doi.org/10.3389/fnhum.2016.00564">giving competitive Scrabble players and a group of Scrabble non-experts a task that was similar to Scrabble but used symbols instead of letters</a>. In that task, Scrabble players were no better than anyone else in terms of their processing speed or accuracy.</p>
<p>We also investigated whether <a href="https://doi.org/10.1016/j.neurobiolaging.2018.05.015">Scrabble expertise protects players from the effects of brain aging</a>. Again, the answer seems to be no. Older Scrabble players still show the normal effects of aging, like slower processing speed.</p>
<figure class="align-center ">
<img alt="A man with grey hair and a woman with grey hair look down at a Scrabble board on a table in front of them." src="https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=413&fit=crop&dpr=1 600w, https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=413&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=413&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=519&fit=crop&dpr=1 754w, https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=519&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/442245/original/file-20220124-21-idxrqm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=519&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">An elderly couple plays Scrabble at their Toronto home in 2013.</span>
<span class="attribution"><span class="source">THE CANADIAN PRESS/Nathan Denette</span></span>
</figcaption>
</figure>
<p>In both Scrabble and Wordle, players need to search their word memory based on letters, shuffle letters across positions to find solutions or plays — the meaning of the words is irrelevant. Because of these similarities, many of the brain processes involved in Scrabble are probably also engaged when solving Wordles. </p>
<p>Our research with people who are not Scrabble experts shows that <a href="https://doi.org/10.3390/e23030304">mental processes start to change quite quickly when people are asked to take on a new word recognition task</a>. That means it’s very likely your Wordle habit has already caused slight changes in the brain processes you use to solve the puzzles. </p>
<p>Those changes help you to play Wordle, but probably don’t help you with anything else.</p>
<h2>Why do some people love puzzles?</h2>
<p>Wordle has become a habit for millions, but for others it’s not appealing. </p>
<p>There are probably lots of reasons for this, but one explanation could be differences in what people find motivating. Some people enjoy puzzles and thinking challenges more than others. This type of motivation is referred to as <a href="https://doi.apa.org/doi/10.1037/0033-2909.119.2.197">need for cognition</a>, and people who have a high need for cognition tend to seek out mental challenges like word games and puzzles.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1484239830464696324"}"></div></p>
<p>In Scrabble, there are usually multiple possible plays that could advance the game, but Worldles have a single right answer. With only one Wordle released per day, everyone is solving the same puzzle. The online game’s sharing options also allow us to share our results with others without giving the answer away.</p>
<p>That means Wordle is also creating an opportunity for shared experience at a time when many people are feeling disconnected from others. A Wordle habit is not likely to make you smarter or ward off brain aging, but it may give you a daily dose of complex cognition combined with social interaction — and that can be a very good thing.</p><img src="https://counter.theconversation.com/content/175227/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Penny Pexman receives funding from the Natural Sciences and Engineering Research Council of Canada. She is a member of the Hotchkiss Brain Institute.</span></em></p>Like a Scrabble habit, a passion for Wordle isn’t likely to make you smarter or ward off brain aging. But it may give you a daily dose of complex cognition combined with social interaction.Penny Pexman, Professor of Psychology, University of CalgaryLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1679002021-10-13T00:06:23Z2021-10-13T00:06:23ZDeciphering the Philosophers’ Stone: how we cracked a 400-year-old alchemical cipher<p>What secret alchemical knowledge could be so important it required sophisticated encryption?</p>
<p>The setting was Amsterdam, 2019. A conference organised by the <a href="https://www.ambix.org/">Society for the History of Alchemy and Chemistry</a> had just concluded at the <a href="https://embassyofthefreemind.com/en/">Embassy of the Free Mind</a>, in a lecture hall opened by historical-fiction author Dan Brown. </p>
<p>At the conference, Science History Institute Postdoctoral Researcher Megan Piorko presented a curious manuscript belonging to English alchemists John Dee (1527–1608) and his son Arthur Dee (1579–1651). In the pre-modern world, alchemy was a means to understand nature through ancient secret knowledge and <a href="https://www.science.org/doi/abs/10.1126/science.332.6032.914">chemical experiment</a>. </p>
<p>Within Dee’s alchemical manuscript was a cipher table, followed by encrypted ciphertext under the heading “Hermeticae Philosophiae medulla” — or Marrow of the Hermetic Philosophy. The table would end up being a valuable tool in decrypting the cipher, but could only be interpreted correctly once the hidden “key” was found. </p>
<p>It was during post-conference drinks in a dimly lit bar that Megan decided to investigate the mysterious alchemical cipher — with the help of her colleague, University of Graz Postdoctoral Researcher Sarah Lang.</p>
<h2>A recipe for the elixir of life</h2>
<p>Megan and Sarah shared their initial analysis on a history of chemistry <a href="https://www.jargonium.com/post/cipher-treasure-hunt">blog</a> and presented the historical discovery to cryptology experts from around the world at the 2021 <a href="https://ecp.ep.liu.se/index.php/histocrypt/issue/view/15">HistoCrypt</a> conference. </p>
<p>Based on the rest of the notebook’s contents, they believed the ciphertext contained a recipe for the fabled Philosophers’ Stone — an elixir that supposedly prolongs the owner’s life and grants the ability to <a href="https://www.scientificamerican.com/article/fact-or-fiction-lead-can-be-turned-into-gold/">produce gold from base metals</a>. </p>
<p>The mysterious cipher received much interest, and Sarah and Megan were soon inundated with emails from would-be code-breakers. That’s when Richard Bean entered the picture. Less than a week after the HistoCrypt proceedings went live, Richard contacted Lang and Piorko with exciting news: he’d cracked the code. </p>
<p>Megan and Sarah’s initial hypothesis was confirmed; the encrypted ciphertext was indeed an alchemical recipe for the Philosophers’ Stone. Together, the trio began to translate and analyse the 177-word passage. </p>
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<strong>
Read more:
<a href="https://theconversation.com/why-the-ancient-promise-of-alchemy-is-fulfilled-in-reading-109497">Why the ancient promise of alchemy is fulfilled in reading</a>
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<h2>The alchemist behind the cipher</h2>
<p>But who wrote this alchemical cipher in the first place, and why encrypt it? </p>
<p>Alchemical knowledge was shrouded in secrecy, as practitioners believed it could only be understood by true adepts.</p>
<p>Encrypting the most valuable trade secret, the Philosophers’ Stone, would have provided an added layer of protection against alchemical fraud and the unenlightened. Alchemists spent their lives searching for this vital substance, with many <a href="https://www.livescience.com/54162-newton-recipe-for-philosophers-stone-rediscovered.html">believing they</a> had the key to successfully unlocking the secret recipe. </p>
<p>Arthur Dee was an English alchemist and spent most of his career as royal physician to Tsar Michael I of Russia. He continued to add to the alchemical manuscript after his father’s death — and the cipher appears to be in Arthur’s handwriting. </p>
<p>We don’t know the exact date John Dee, Arthur’s father, started writing in this manuscript, or when Arthur added the cipher table and encrypted text he titled “The Marrow of Hermetic Philosophy”. </p>
<p>However, we do know Arthur wrote another manuscript in 1634 titled <em>Arca Arcanorum</em> — or Secret of Secrets — where he celebrates his alchemical success with the Philosophers’ Stone, claiming he discovered the true recipe. </p>
<p>He decorated <em>Arca Arcanorum</em> with an emblem copied from a <a href="https://hdl.huntington.org/digital/iiif/p15150coll7/30526/full/full/0/default.jpg">medieval alchemical scroll</a>, illustrating the allegorical process of alchemical transmutation necessary for the Philosophers’ Stone.</p>
<h2>Cracking the code</h2>
<p>What clues led to decrypting the mysterious Marrow of the Hermetic Philosophy passage? </p>
<p>Adjacent to the encrypted text is a table resembling one used in a traditional style of cipher called a <a href="https://www.cryptogram.org/downloads/aca.info/ciphers/Porta.pdf">Bellaso/Della Porta cipher</a> — invented in 1553 by Italian cryptologist Giovan Battista Bellaso, and written about in 1563 by Giambattista della Porta. This was the first clue.</p>
<p>The Latin title indicated the text itself was also in Latin. This was corroborated by the lack of letters V and J in the cipher table, as V and J are interchangeable with U and I, respectively, in printed Latin text.</p>
<p>This was good news, as Richard had access to Latin statistical models from <a href="https://doi.org/10.3384/ecp2020171005">previous decryption projects</a>. Armed with this information, he set off in search of patterns that would lead him to the cipher “key” — a word or phrase that could be used in conjunction with the cipher table to decipher the text.</p>
<figure class="align-center ">
<img alt="Cipher table" src="https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=497&fit=crop&dpr=1 600w, https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=497&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=497&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=625&fit=crop&dpr=1 754w, https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=625&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/423788/original/file-20210929-13-awmjtw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=625&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">An encryption table for the Bellaso / Della Porta cipher, invented in Italy in 1553. Only ten rows are shown, as wx / yz were not in the key.</span>
</figcaption>
</figure>
<p>Richard soon realised the key was included at the end of the text, which is unusual. It was surprisingly long too, made up of 45 letters — arduous even for today’s computer-password standards. The trio would later realise the key was also written elsewhere in the manuscript, hidden in plain sight. </p>
<figure class="align-right ">
<img alt="Latin and enciphered text written in notebook." src="https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/424328/original/file-20211003-44779-g38qqa.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Richard found the key and used it, along with the cipher table, to decrypt the cipher.</span>
<span class="attribution"><span class="source">Author provided</span></span>
</figcaption>
</figure>
<p>In keeping with the typical encryption practices of the period, Arthur Dee had written the key on the back of the cipher table. It read: <em>sic alter iason aurea felici portabis uellera colcho</em>, meaning “like a new Jason you will carry the Golden Fleece away from the lucky Colchian”. </p>
<h2>An ancient myth</h2>
<p>This key is adapted from the last verses of an alchemical poem by Giovanni Aurelio Augurello titled <em>Chrysopoeia</em> (circa 1505), with “chrysopoeia” also being the ancient Greek word for the art of gold-making.</p>
<p>The poem is about the ancient Greek myth of Jason and the Argonauts, which was reinterpreted during the early modern period as an allegory for alchemy. In the myth of Jason and the Argonauts, the Argonauts sail to the land of Colchis (in modern-day Georgia) to retrieve the “Golden Fleece”. In an alchemical context, the fleece is a symbol for the Philosophers’ Stone. </p>
<p>The actual text of the Marrow of the Hermetic Philosophy mentions taking an alchemical “egg” — not further described — from an <a href="https://my.matterport.com/show/?m=f46fDRkYp8q">athanor</a>, which is a type of furnace used for gentle heating over a long period of time.</p>
<p>Afterwards, instructions are given for how long to wait until the different alchemical phases ensue (the blackening, whitening and the red phase). It says the end product — either a silver tincture or the gold-making elixir — will depend on when the process is stopped.</p>
<p>If the directions are followed correctly, the code-cracking reader is promised:</p>
<blockquote>
<p>… then you will have a truly gold-making elixir by whose benevolence all the misery of poverty is put to flight and those who suffer from any illness will be restored to health.</p>
</blockquote>
<p>Contrary to what was believed for a long time, alchemical recipes do contain chemical processes which can be reproduced in modern laboratories. It’s only towards the end (during the production of the Philosophers’ Stone) that the recipe becomes too vague to reproduce — at least not without further interpretation.</p>
<p>However, they do sometimes produce a blood-red glass (which is what the stone was said to look like). </p>
<h2>Journey to the centre of the archive</h2>
<p>What can we learn from historical ciphers? Cryptology experts have just scratched the surface of early-modern encryption practices. Much secret alchemical knowledge remains uncovered from a time when making gold and extending the natural limit of life was believed possible through alchemy. </p>
<p>The decryption of this 400-year-old cipher suggests we have much ground to dig through yet. Who knows what other alchemical ciphers are waiting to be discovered in the depths of the archive?</p>
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<strong>
Read more:
<a href="https://theconversation.com/declassified-cold-war-code-breaking-manual-has-lessons-for-solving-impossible-puzzles-161595">Declassified Cold War code-breaking manual has lessons for solving 'impossible' puzzles</a>
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<img src="https://counter.theconversation.com/content/167900/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Megan Piorko receives funding from the Science History Institute. </span></em></p><p class="fine-print"><em><span>Sarah Lang receives funding from the Science History Institute and the University of Graz.</span></em></p><p class="fine-print"><em><span>Richard Bean does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The secret recipe came from Arthur Dee, 17th-century alchemist and royal physician to the Tsar.Richard Bean, Research Fellow, The University of QueenslandMegan Piorko, Allington Postdoctoral Fellow, Science History Institute, Georgia State UniversitySarah Lang, PostDoc in Digital Humanities at Centre of Information Modelling (University of Graz), University of GrazLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1615952021-05-30T20:06:31Z2021-05-30T20:06:31ZDeclassified Cold War code-breaking manual has lessons for solving ‘impossible’ puzzles<figure><img src="https://images.theconversation.com/files/403013/original/file-20210526-16-b39162.png?ixlib=rb-1.1.0&rect=437%2C2%2C1006%2C549&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.nsa.gov/Portals/70/documents/news-features/declassified-documents/cryptologic-spectrum/Callimahos_Course.pdf">NSA</a></span></figcaption></figure><p>The <a href="https://www.nsa.gov/">United States National Security Agency</a> — the country’s premier signals intelligence organisation — recently declassified a Cold War-era document about code-breaking.</p>
<p>The <a href="https://www.governmentattic.org/39docs/NSAmilitaryCryptalyticsPt3_1977.pdf">1977 book</a>, written by cryptologist <a href="https://en.wikipedia.org/wiki/Lambros_D._Callimahos">Lambros Callimahos</a>, is the last in a trilogy called Military Cryptanalytics. It’s significant in the history of cryptography, as it explains how to break all types of codes, including military codes, or puzzles — which are created solely for the purpose of a challenge.</p>
<p>The first two parts of the trilogy were <a href="https://en.wikipedia.org/wiki/Military_Cryptanalytics">published publicly in the 1980s</a> and covered solving well-known types of <a href="https://en.wikipedia.org/wiki/Classical_cipher">classical cipher</a>. </p>
<p>But in 1992, the US Justice Department claimed releasing the third book could harm national security by revealing the NSA’s “<a href="https://www.nytimes.com/1992/11/28/us/in-retreat-us-spy-agency-shrugs-at-found-secret-data.html">code-breaking prowess</a>”. It was finally released in December last year. </p>
<figure class="align-center ">
<img alt="Man wearing bow tie" src="https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/402995/original/file-20210526-17-1e112f3.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Lambros D. Callimahos, the author of Military Cryptanalytics.</span>
<span class="attribution"><span class="source">NSA</span></span>
</figcaption>
</figure>
<h2>Lessons for code-breakers</h2>
<p>A key part of Callimahos’s book is a chapter titled Principles of Cryptodiagnosis, which describes a systematic three-step approach to solving a message encrypted using an unknown method. </p>
<p>An intelligence agency might intercept thousands of messages made in a target country’s ciphers, in which case they already know the method. But if they encounter something new, they must first and foremost figure out the encryption method, or risk wasting time.</p>
<p>As Callimahos details in his chapter, the code-breaker must begin with all the necessary data. This includes the ciphertext (the enciphered text hiding the real message), any known underlying plaintext (text from before the encryption was applied), as well as important contextual information.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/cryptology-from-the-crypt-how-i-cracked-a-70-year-old-coded-message-from-beyond-the-grave-122465">Cryptology from the crypt: how I cracked a 70-year-old coded message from beyond the grave</a>
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<p>For puzzles, part of the plaintext may be given to help the solver. With confidential military messages, the solver may suspect certain words have been encoded into the ciphertext, based on past knowledge. For example, there may be key terms such as “message begins”, “message ends” or “secret”, or specific names, places or addresses.</p>
<p>The code-breaker then arranges and rearranges the data to find non-random characteristics. After this, they can recognise and explain these characteristics. In other words, they’ve found the cipher method.</p>
<p>Applying these steps is an example of “<a href="https://en.wikipedia.org/wiki/Bayesian_inference">Bayesian inference</a>”. The code-breaker considers the <a href="https://www.cs.tufts.edu/%7Enr/cs257/archive/jack-good/weight-of-evidence.pdf">weight of evidence</a> and guesses the likely <a href="https://www.gchq.gov.uk/information/codes-chess-and-kubrick-life-jack-good">cause of an observed effect</a>. </p>
<h2>The Zodiac and Kryptos ciphers</h2>
<p>Last year, the famous 1969 <a href="https://en.wikipedia.org/wiki/Zodiac_Killer">Zodiac killer cipher</a>, known as Z340, was <a href="https://www.abc.net.au/news/2020-12-12/zodiac-killer-code-cracked-by-australian-mathematician/12977342">solved</a> by an international team of code-breakers after 51 years. The team carefully and systematically developed a <a href="http://zodiackillerciphers.com/wiki/index.php?title=Encyclopedia_of_observations">list of observations</a> over many years.</p>
<p>Using a process called <a href="https://en.wikipedia.org/wiki/Monte_Carlo_method">Monte Carlo sampling</a>, they <a href="https://youtu.be/iuNyQ44JYxM">tested</a> whether the patterns observed in the ciphertext were random or not. Together with a detailed knowledge of <a href="https://youtu.be/3Hk__Hk5c9M">the context of the cipher</a> and a solution for a <a href="http://zodiackillerciphers.com/408/key.html">previous cipher by the Zodiac killer</a>, they correctly guessed the encryption method used. </p>
<p>One of the Zodiac cipher solvers, David Oranchak, <a href="https://eng.vt.edu/magazine/stories/spring-2020/the-new-cryptographers.html">said</a> in his opinion it was “at about a seven or eight out of ten in difficulty to decipher”. </p>
<p>Similarly, US artist <a href="https://en.wikipedia.org/wiki/Jim_Sanborn">Jim Sanborn’s</a> famous Kryptos sculpture, located at the Central Intelligence Agency, has long confounded attempts to unlock its code. It contains four encrypted passages to challenge the agency’s employees. The final passage, known as K4, remains unsolved after 30 years.</p>
<figure class="align-center ">
<img alt="Kryptos sculpture at CIA" src="https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=759&fit=crop&dpr=1 600w, https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=759&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=759&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=954&fit=crop&dpr=1 754w, https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=954&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/402831/original/file-20210526-13-1gl3gk9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=954&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Kryptos sculpture at the CIA. K4 is visible in the last three lines on the right; [NYP]VTTMZFPK is said to read [BER]LINCLOCK in plaintext.</span>
<span class="attribution"><span class="source">Carol Highsmith, Library of Congress</span></span>
</figcaption>
</figure>
<p>When Kryptos’s code designer Ed Scheidt was asked to rate the cipher’s difficulty, he estimated it as being around a nine out of ten on the same scale. He said his intention was for it to be solved in <a href="https://www.baltimoresun.com/news/bs-xpm-2000-03-17-0003180448-story.html">five</a>, <a href="https://www.wired.com/2009/04/ff-kryptos/">seven</a> or maybe <a href="https://www.defcon.org/images/defcon-12/dc-12-presentations/Dunin/dc-12-dunin.ppt">ten</a> years.</p>
<p>So what has made K4 so difficult? For one, with only 97 letters the passage is very short, meaning less data and fewer clues. The enciphering method used to create it is unknown, and there’s little context as to how it may have been enciphered. </p>
<p>One classic book on mathematical problem solving, How to Solve It by George Pólya, <a href="https://math.hawaii.edu/home/pdf/putnam/PolyaHowToSolveIt.pdf">suggests</a> a general principle for solving any problem is to refer to a similar problem that has already been solved. This principle applies in the historical puzzle world, too. </p>
<p>However, Scheidt also noted there was a “change in the methodology” as the Kryptos message progressed — done intentionally to make it increasingly difficult. </p>
<p>It could also be that Sanborn accidentally introduced an error in K4 during the construction of the Kryptos sculpture, which would mean solvers are wasting their time. Making <a href="https://www.nytimes.com/2006/04/22/us/a-break-for-code-breakers-on-a-cia-mystery.html">a mistake</a> during enciphering can render a puzzle impossible to solve. In such cases, the creator should admit this to prospective code-breakers.</p>
<h2>Lessons for code-makers</h2>
<p>Looking at a puzzle from the code-maker’s perspective is important. A skilled code-maker should leave at least some non-random patterns in the cipher, so as to not make their puzzle impossible.</p>
<p>Imagine you’ve created a puzzle, but after many years your intended audience has failed to solve it. If you still want it solved, you have to <a href="https://www.nytimes.com/2014/11/21/us/another-kryptos-clue-is-offered-in-a-24-year-old-mystery-at-the-cia.html">start releasing clues</a>. Some puzzles, such as the 1979 book <a href="https://en.wikipedia.org/wiki/Masquerade_(book)">Masquerade</a> and the <a href="https://youtu.be/meaUE2b5whI">Decipher Puzzles</a>, were only solved after clues were released. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=793&fit=crop&dpr=1 600w, https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=793&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=793&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=996&fit=crop&dpr=1 754w, https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=996&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/402996/original/file-20210526-13-i4f1yr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=996&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Masquerade by Kit Williams was a 1979 puzzle book.</span>
<span class="attribution"><span class="source">Wikimedia</span></span>
</figcaption>
</figure>
<p>However, if nobody has solved your puzzle even after you release <a href="https://twitter.com/jswatz/status/1297658577914667008">many clues</a>, then the code is simply too tough to crack.</p>
<p>Cryptographer Helen Fouché Gaines wrote about this in her <a href="https://archive.org/details/1956Cryptanalysis-AStudyOfCiphersAndTheirSolution">1939 book</a>. The creator of such a puzzle, she said, “fails to submit material in proportion to the amount of complication he has introduced”. </p>
<p>This means you may have to eventually reveal the method you used. One example is a complex algorithm known as <a href="https://en.wikipedia.org/wiki/Chaocipher">Chaocipher</a>. While Chaocipher messages were designed to be highly difficult, they’re virtually impossible to decipher without knowing the method.</p>
<p>A <a href="https://www.nsa.gov/Portals/70/documents/news-features/declassified-documents/cia-kryptos-sculpture/KRYPTOS_Summary.pdf">2007 NSA presentation</a> about Kryptos mentions how “dozens” of agency staff have failed to solve K4. But as more historical texts become declassified and our computational, storage and networking capacity grows, perhaps one day an amateur code-breaker — and not an agent of the NSA — will crack the elusive passage.</p>
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Read more:
<a href="https://theconversation.com/how-hard-is-it-to-scramble-rubiks-cube-129916">How hard is it to scramble Rubik’s Cube?</a>
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<img src="https://counter.theconversation.com/content/161595/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Richard Bean does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Researchers tried several times to have the document declassified, including in 1992, 2004 and 2016. It was initially written to help American NSA agents crack difficult coded messages.Richard Bean, Research Fellow, The University of QueenslandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1299162020-01-31T03:24:43Z2020-01-31T03:24:43ZHow hard is it to scramble Rubik’s Cube?<figure><img src="https://images.theconversation.com/files/312387/original/file-20200129-92992-aoyptf.jpg?ixlib=rb-1.1.0&rect=43%2C0%2C4896%2C3261&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">What a scramble.</span> <span class="attribution"><span class="source">Shutterstock.com</span></span></figcaption></figure><p>Rubik’s Cube has been one of the world’s favourite puzzles for 40 years. Several different methods have been devised for solving it, as explained in countless books. Expert “speedcubers” can solve it in a matter of <a href="https://www.worldcubeassociation.org">seconds</a>.</p>
<p>In addition to such feats of astounding dexterity, there are many fascinating mathematical questions related to Rubik’s Cube. A move of the cube consists of rotating one of the six faces by either 90, 180, or 270 degrees. A staggering 43,252,003,274,489,856,000 possible states can be obtained by applying sequences of moves to the solved state. </p>
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Read more:
<a href="https://theconversation.com/how-to-solve-a-rubiks-cube-in-five-seconds-51359">How to solve a Rubik's cube in five seconds</a>
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<p>Despite this complexity, it was <a href="http://www.cube20.org">shown</a> in 2010 that Rubik’s Cube can always be solved in 20 moves or fewer, regardless of the initial state. This number is referred to as “God’s number”, as all known solution methods used by humans typically use significantly more moves than this optimal value.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/310943/original/file-20200120-69551-11mh88v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Rubik’s Cube in the solved state.</span>
<span class="attribution"><span class="source">Mike Gonzalez (TheCoffee)</span></span>
</figcaption>
</figure>
<p>But what about the opposite question: how many moves are required to scramble a solved cube? At first glance, this sounds like a much easier question than computing God’s number. After all, unlike solving a cube, scrambling one takes no skill whatsoever. </p>
<p>Similar questions have been answered successfully for card shuffling. A famous example is the <a href="https://www.nytimes.com/1990/01/09/science/in-shuffling-cards-7-is-winning-number.html">1990 study</a> of the “<a href="https://en.wikipedia.org/wiki/Shuffling#Riffle">riffle shuffle</a>” by mathematicians Dave Bayer and Perci Diaconis. A deck of cards is defined as “mixed” if its ordering is random, with each possible order having the same probability of appearing. Bayer and Diaconis showed that seven riffle shuffles are necessary and sufficient to approximately mix a standard deck of playing cards.</p>
<p>Last year, mathematicians published a similar study of the <a href="https://www.quantamagazine.org/mathematicians-calculate-how-randomness-creeps-in-20191112/">15 puzzle</a>, which consists of a 4x4 square filled with 15 sliding tiles and one empty space.</p>
<h2>What does it mean for a cube to be scrambled?</h2>
<p>A typical person trying to scramble a Rubik’s Cube would repeatedly perform random moves on it. The resulting random sequence of states is a special case of what mathematicians call a <a href="https://www.britannica.com/science/Markov-process">Markov chain</a>. The key property is that given the current state, the probability of what the next state will be does not depend on any of the previous states.</p>
<p>Applying the theory of Markov chains to cube scrambling, it follows that as the number of random moves increases, the probability of being in any particular one of the possible states becomes closer and closer to 1/43,252,003,274,489,856,000. Mathematicians call this a “uniform probability distribution”, as each possible state occurs with the same probability. </p>
<p>After any given number of random moves, the state of the cube will be random, but its probability distribution will not be exactly uniform; some states will be more likely to occur than others.</p>
<p>Let <em>d(t)</em> describe how much the probability distribution after <em>t</em> random moves differs from the uniform probability distribution. As the number of random moves (<em>t</em>) increases, the value of <em>d(t)</em> will decrease. The cube being scrambled corresponds to <em>d(t)</em> being small.</p>
<h2>Markov-chain Monte Carlo</h2>
<p>In the theory of Markov chains, this decrease in <em>d(t)</em> is called “mixing”. Besides card shuffling and puzzle scrambling, the theory of Markov chain mixing also has very serious practical applications. One of the most important computational tools in modern science and engineering is the Monte Carlo method. This method, like the famous casino after which it is named, relies fundamentally on chance. In essence, it attempts to approximately solve hard mathematical problems using multiple random guesses. </p>
<p>In practice, Markov chains are often used to produce these random states. To understand the accuracy of these Markov-chain Monte Carlo methods, the key task is to estimate how quickly <em>d(t)</em> decreases as <em>t</em> increases.</p>
<h2>The pocket cube</h2>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/311011/original/file-20200120-69531-2ehuze.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Pocket cube in a scrambled state.</span>
<span class="attribution"><span class="source">Mike Gonzalez (TheCoffee)</span></span>
</figcaption>
</figure>
<p>Studying the scrambling problem for the standard 3x3x3 Rubik’s Cube is currently a fascinating unsolved challenge. However, it becomes quite manageable if we turn our attention to a smaller 2x2x2 version, called the pocket cube. </p>
<p>In this cube, the edge and centre pieces are absent and only the corner pieces remain. The pocket cube has only 3,674,160 possible states, and its God’s number is only 11. </p>
<p>In the graph below, we plot <em>d(t)</em> for the pocket cube. After 11 moves, <em>d(t)</em> is still very large, at 0.695. The first value of <em>t</em> that yields a <em>d(t)</em> value below 0.25 (often called “the mixing time” in Markov chain theory) is 19. After 25 moves <em>d(t)</em> is 0.092; after 50 moves it is 0.0012; and after 100 moves it is 0.00000017. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=280&fit=crop&dpr=1 600w, https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=280&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=280&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=352&fit=crop&dpr=1 754w, https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=352&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/311005/original/file-20200120-69551-19eg22o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=352&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Distance of the pocket cube distribution from uniform after t moves.</span>
<span class="attribution"><span class="source">Eric Zhou</span></span>
</figcaption>
</figure>
<p>So how many moves should you use to fully scramble a pocket cube? The answer depends on how small you would like <em>d(t)</em> to be. However, it is certainly true that God’s number of moves is insufficient. As a bare minimum, one should not use fewer than 19 moves. Further details, including code to compute <em>d(t)</em>, are available <a href="https://github.com/Peakergzf/pocket-cube-mixing">here</a>.</p>
<p>And of course, once you’ve scrambled your cube, all that’s left to do is solve it again.</p>
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Read more:
<a href="https://theconversation.com/your-guide-to-solving-the-next-online-viral-maths-problem-75303">Your guide to solving the next online viral maths problem</a>
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<img src="https://counter.theconversation.com/content/129916/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Tim Garoni is a Chief Investigator of the Australian Research Council Centre of Excellence in Mathematical and Statistical Frontiers (ACEMS)</span></em></p><p class="fine-print"><em><span>Zongzheng Zhou is a Research Fellow at the Australian Research Council Centre of Excellence in Mathematical and Statistical Frontiers (ACEMS).</span></em></p><p class="fine-print"><em><span>Peaker Guo does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Scrambling it is much easier than solving it. But it still involves some fascinating questions, such as the number of random moves needed to consider the cube truly messed up.Tim Garoni, Associate Professor, School of Mathematics, Monash UniversityPeaker Guo, Honours student, Faculty of IT, Monash UniversityZongzheng Zhou, Research fellow, School of Mathematics, Monash UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1248912019-10-22T18:58:07Z2019-10-22T18:58:07ZYour brain approaches tricky tasks in a surprisingly simple way<figure><img src="https://images.theconversation.com/files/298045/original/file-20191022-28100-1p7sz04.jpg?ixlib=rb-1.1.0&rect=10%2C110%2C6699%2C4134&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">It gets easier with practice.</span> <span class="attribution"><span class="source">Duntrune Studios/Shutterstock</span></span></figcaption></figure><p>Have you ever sat down to complete your morning crossword or Sudoku and wondered about what’s happening in your brain? Somewhere in the activity of the billions of neurons in your brain lies the code that lets you remember a key word, or apply the logic required to complete the puzzle. </p>
<p>Given the brain’s intricacy, you might assume that these patterns are incredibly complex and unique to each task. But <a href="https://www.nature.com/articles/s41593-018-0312-0">recent research</a> suggests things are actually more straightforward than that.</p>
<p>It turns out that many structures in your brain work together in precise ways to coordinate their activity, shaping their actions to the requirements of whatever it is that you’re trying to achieve. </p>
<p>We call these coordinated patterns the “low-dimensional manifold”, which you can think of as analogous to the major roadways that you use to commute to and from work. The majority of the traffic flows along these major highways, which represent an efficient and effective way to get from A to B. </p>
<p>We have found evidence that most brain activity follows these types of patterns. In very simple terms, this saves your brain from needing to work everything out from scratch when performing a task. If someone throws you a ball, for instance, the low-dimensional manifold allows your brain to swiftly coordinate the muscle movements needed to catch the ball, rather than your brain needing to learn how to catch a ball afresh each time.</p>
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Read more:
<a href="https://theconversation.com/how-the-brain-prepares-for-movement-and-actions-111674">How the brain prepares for movement and actions</a>
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<p>In a study <a href="https://www.cell.com/neuron/fulltext/S0896-6273(19)30775-5">published today in the journal Neuron</a>, my colleagues and I investigated these patterns further. Specifically, we wanted to find out whether they play a role in shaping brain activity during really challenging cognitive tasks that require lots of concentration. </p>
<p>We scanned people’s brains with high-resolution functional magnetic resonance imaging (fMRI) while they performed a <a href="https://en.wikipedia.org/wiki/Latin_square">Latin squares task</a>, which is similar to a Sudoku puzzle but uses shapes instead of numbers. Anyone who has played Sudoku before their morning coffee knows how much focus and concentration is required to solve it. </p>
<p>The idea behind the Latin squares task is to identify the missing shape in a particular location in a grid, given that each shape can only show up once in each row and column. We created three different levels of difficulty, defined by how many different rows and columns needed to be inspected to arrive at the correct answer. </p>
<h2>Directing traffic</h2>
<p>Our prediction was that performing the more difficult versions of the task would lead to a reconfiguration of the low-dimensional manifold. To return to the highway analogy, a tricky task might pull some brain activity off the highway and onto the back streets to help get around the congestion.</p>
<p>Our results confirmed our predictions. More difficult trials showed different patterns of brain activation to easy ones, as if the brain’s traffic was being rerouted along different roads. The trickier the task, the more the patterns changed. </p>
<p>What’s more, we also found a link between these changed brain activation patterns and the increased likelihood of making a mistake on the harder version of the Latin Squares test. </p>
<p>In a way, attempting a difficult task is like trying out a new rat run on your morning commute – you might succeed, but in your haste and stress you might also be more likely to take a wrong turn.</p>
<p>Overall, these results suggests that our brain activity perhaps isn’t as complicated as we once thought. Most of the time, our brain is directing traffic along pretty well-established routes, and even when it needs to get creative it is still trying to send the traffic to the same ultimate destination.</p>
<p>This leaves us with an important question: how does the brain achieve this level of coordination? </p>
<p>One possibility is that this function is fulfilled by the <a href="https://www.britannica.com/science/thalamus">thalamus</a>, a structure that lies deep in the brain but is connected to almost the entire rest of the brain. </p>
<p>Importantly, the circuitry of the thalamus is such that it can act as a filter for ongoing activity in the cerebral cortex, the brain’s main information processing centre, and therefore could exert the kind of influence we were looking for.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/298044/original/file-20191022-28112-nv7utl.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Positions of the thalamus and the cerebral cortex within the brain.</span>
<span class="attribution"><span class="source">Pikovit/Shutterstock</span></span>
</figcaption>
</figure>
<p>Patterns of activity in the thalamus are hard to decipher in traditional neuroimaging experiments. But fortunately, the <a href="https://cai.centre.uq.edu.au/facilities/human-imaging/7t-magnetom">high-resolution MRI scanner used in our study</a> collected by my colleagues Luca Cocchi and Luke Hearne allowed us to observe them in detail.</p>
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Read more:
<a href="https://theconversation.com/neuroscience-in-pictures-the-best-images-of-the-year-89077">Neuroscience in pictures: the best images of the year</a>
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<p>Sure enough, we saw a clear link between activity in the thalamus and the flow of activity in the low-dimensional manifold. This suggests that when performing particular tasks, the thalamus helps to shape and constrain the activity in the cortex, a bit like a police officer directing busy traffic. </p>
<p>So next time you sit down to play Sudoku, spare a thought for your thalamus, and the low-dimensional manifold that it helps to create. Together, they’re shaping the brain activity that will ultimately help you solve the puzzle.</p><img src="https://counter.theconversation.com/content/124891/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>James Shine receives funding from the National Health and Medical Research Council. He is affiliated with The University of Sydney and the Organisation for Human Brain Mapping Australia.</span></em></p>Despite its huge complexity, your brain directs its neural traffic in relatively straightforward ways when approaching cognitively demanding tasks such as puzzles.James Shine, Robinson Fellow, University of SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1084482018-12-11T15:34:42Z2018-12-11T15:34:42ZDoes problem-solving really protect against cognitive decline in old age?<figure><img src="https://images.theconversation.com/files/249699/original/file-20181210-76959-12gdh90.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/download/confirm/636441224?src=iUS8QHdAkyyca5dW5RW0FQ-1-15&size=medium_jpg">adriaticfoto/Shutterstock</a></span></figcaption></figure><p>“Use it or lose it” is the received wisdom when it comes to cognitive ability. But is there any truth in this old saw? Our <a href="https://www.bmj.com/content/363/bmj.k4925">latest study</a> suggests that it depends how much “it” you have to start with.</p>
<p>Previous <a href="https://www.ncbi.nlm.nih.gov/pubmed/18179299">observational studies</a> that looked at the effect of doing mentally stimulating activities, such as puzzles, on cognitive ability have largely supported the “use it or lose it” hypothesis. However, these studies have often been based on snapshots in time – so-called cross-sectional studies. To find out if there really is a link between mental engagement over a lifetime and cognitive ability in old age, you need to track people’s habits and mental abilities over a lifetime.</p>
<p>For our study, reported in The BMJ, we wanted to know if mental engagement protects against cognitive decline, or if those with a cognitive advantage engage more, giving the impression that this type of behaviour is responsible for their superior abilities. To answer these questions, we needed to work closely with our study participants and measure their intellectual abilities repeatedly over time and compare their abilities to their performance in early life.</p>
<p>Scotland is unique because in 1947 almost all 11-year-old children took the same mental ability test. The Scottish Council for Research in Education preserved these records and in 1998 allowed us to contact the surviving people who took the test.</p>
<p>We tested people living independently without dementia on up to five occasions over 15 years. Demographic, clinical, questionnaire and psychological data were recorded at all assessments and related to changes in performance on repeated verbal-memory and mental-speed tests.</p>
<h2>Notable results</h2>
<p>Our results are notable because they include childhood intelligence data from a rare historical survey of intelligence. They showed that, in late life, levels of mental ability are strongly linked to current levels of engagement in problem-solving.</p>
<p>Our study was able to account for childhood intelligence and education and revealed the rate of decline in cognition in late life did not differ between people who reported different levels of engagement. However, levels of engagement were associated with the performance at entry, aged 64. </p>
<p>Childhood intelligence was associated with intellectual engagement, which raises the question: do smarter people engage more, or are they smarter because they engage? If the latter were true in late life, then we would expect some influence on the rate of decline.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/249939/original/file-20181211-76974-1b4uqrw.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Boosting resilience against cognitive decline.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/download/confirm/1240427278?src=yMdtsy_4eobaHkToJuTQQA-1-21&size=medium_jpg">furtseff/Shutterstock</a></span>
</figcaption>
</figure>
<p>We have shown that intellectual engagement in this group of people without dementia is not related to later rates of cognitive decline. But engagement is linked to intellectual gain acquired from childhood to late middle age when we first started testing them. In other words, doing puzzles and other intellectually engaging things over a lifetime improves your IQ, so that when the inevitable cognitive decline sets in later life, you have a higher point to start from. The rate of decline is the same for all, regardless of the levels of engagement.</p>
<p>At this stage, we can foresee how lifelong intellectual engagement contributes to protection from falling below some intellectual threshold where you would be considered impaired. This is achieved by starting from a higher point. </p>
<p>Our findings are consistent with <a href="https://www.nejm.org/doi/full/10.1056/NEJMoa022252">comparable studies</a> that followed older people from age 50. We have identified problem-solving as being of specific importance. This suggests that interventions to boost resilience to ageing should include problem-solving components, such as reading complex novels, solving crossword puzzles and practising a musical instrument.</p><img src="https://counter.theconversation.com/content/108448/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Michael Hogan receives funding from the Health Research Board (Ireland), and Horizon 2020 (EU Funding) for a variety of basic and applied research and project work. </span></em></p><p class="fine-print"><em><span>Lawrence Whalley and Roger Staff do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>A new study suggests that being intellectually engaged does nothing to slow cognitive decline, but it does start the decline from a higher point.Roger Staff, Honorary Senior Lecturer in Ageing, University of AberdeenLawrence Whalley, Emeritus Professor of Mental Health, University of AberdeenMichael Hogan, Senior Lecturer, Psychology, University of GalwayLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/948502018-04-19T11:17:13Z2018-04-19T11:17:13ZA playful approach to learning means more imagination and exploration<figure><img src="https://images.theconversation.com/files/215389/original/file-20180418-163962-14n971.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/kid-playing-hopscotch-on-playground-outdoors-332743895?src=uRJ1tIoCFJivQroMqyu02Q-1-43">Shutterstock</a></span></figcaption></figure><p>Play in education is controversial. Although it is widely accepted that very young children need to play, as they progress through the school system, the focus moves quickly to measuring learning. And despite the fact that play is <a href="https://pdfs.semanticscholar.org/57c1/6f405b49e6fa4e38e1725310902bb870ace8.pdf">beneficial throughout life</a>, supporting creativity and happiness, it is still seen by many in education as a frivolous waste of time, and not really relevant to proper learning. </p>
<p>In a recent <a href="https://www.tes.com/news/school-news/breaking-views/tes-talks-toanthony-pellegrini">article in the Times Educational Supplement</a>, educational psychologist Anthony Pellegrini argued that play in schools should be limited to the playground rather than the classroom. Play, by his definition, is carried out for its own sake (or “intrinsically-motivated”) and focused on an experience rather than an outcome (“process-orientated”). </p>
<p>Play should not be for the classroom, he contends, because school activities are teacher led, and therefore not intrinsically motivated. They are focused on an end product – the acquisition of skills and knowledge – rather than a process. </p>
<p>There is a great deal of common sense in what Pellegrini says. He highlights the distinction between the idea of play and simply making lessons fun by using games or toys. He believes there is a danger in using the latter approach solely to motivate learners because the motivation does not come from within. </p>
<p>His call for caution in the wholesale use of play in education, and a push for increased focus on learner motivation is also inherently sensible. However, his creation of a false binary between “play is good” versus “play is bad” is reductionist in the extreme. Most teachers recognise that the use of play, in its many forms, is far more nuanced. </p>
<p>Also, the conclusion that play has no place in the classroom is based on certain assumptions about the state of formal school education in the UK – namely, that it is currently fit for purpose. It accepts that a system of teacher-led, externally dictated curricula and regularly assessed learning is the best way to manage an education system and engage young people in meaningful learning.</p>
<p>But what if education was different? </p>
<p>Imagine instead a classroom where learning is dictated by students’ interests, exploration, curiosity and experimentation. Where learners work together to answer questions that are relevant and fascinating to them. Where they don’t expect to get everything right first time, but can learn from their failures, and delight in their discoveries. Where learning is based on skills and values, far beyond the limitations of testable knowledge, and evaluated by application rather than meaningless tests. Isn’t this, by definition, play?</p>
<p>But we live in the real world. Given the current focus (in the UK education system at least) on set content, memorisation and testing, play is evidently not suited to getting the best results in that context. And while there are notable examples of embedded play in schools, such as <a href="http://www.q2l.org/">Quest to Learn</a> in New York, a fundamental rethink of our current education system is unlikely in the near future. But that doesn’t necessarily mean that play must be confined to the playground. </p>
<h2>Thinking outside the ticked box</h2>
<p>There is an important distinction between play as an activity, and playfulness as an attitude. Playfulness is about being open to new experiences. It’s about imagining, a spirit of make-believe, exploring possibilities. </p>
<p>Playful learning approaches encourage the development of playful learners not just through the use of toys and games, or even play-based teaching approaches, but through the development of fundamental <a href="https://www.tandfonline.com/doi/abs/10.1080/21594937.2017.1382997">playful values</a>. Playfulness is key to creating spaces for positive failure, something that the current education system ignores, with its relentless regime of <a href="https://www.parliament.uk/business/committees/committees-a-z/commons-select/education-committee/news-parliament-2015/primary-assessment-report-published-16-17/">high-stakes testing</a> from an early age.</p>
<p>Many teachers fully appreciate the value of playful thinking but the school system doesn’t make it easy to support the change in mindset that playfulness engenders. They are limited by targets, tests, and inspections. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=300&fit=crop&dpr=1 600w, https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=300&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=300&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=377&fit=crop&dpr=1 754w, https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=377&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/215406/original/file-20180418-163978-14b9hh0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=377&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Splashiness.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/feet-child-yellow-rubber-boots-jumping-668307886?src=NKpA0iOSXYpYy1LB1jgaDQ-1-16">Shutterstock</a></span>
</figcaption>
</figure>
<p>Moving from a focus on play to playfulness, however, offers opportunities to rethink possibilities for school learning, even within this restrictive framework. The <a href="http://eduscapes.playthinklearn.net/">EduScapes</a> project at Manchester Metropolitan University, for example, has developed an approach to learning through collaborative design of an “escape room”. </p>
<p>An escape room is a game form where groups of players work together in the same physical space over a restricted time period to solve a variety of physical, mental and collaborative puzzles and achieve some overall objective (usually escaping the room). </p>
<p>Escape room design offers the opportunity for students to design something playful together, creating a product that can be tested but is not formally assessed. The group based design process aims to support productive failure through iterative testing, develop creativity and problem solving skills, and engender a spirit of play in students and staff alike.</p>
<p>This approach has been used for three years with sixth form students at Cheadle Hulme High School in Stockport as an annual enrichment project. David Woolley, the assistant headteacher, is a strong supporter of the project, which he describes as a “fantastic vehicle to build confidence, develop team working, problem solving, critical thinking and presentation skills in a truly fun and interactive way”. </p>
<p>This project shows that playful learning approaches are possible, and can be effective, but they have to be thoughtfully embedded within the constraints of the current education system. Play is not the problem, but playfulness might just be the answer. At a fundamental level, we need to rethink the systems that constrain teachers from being effective and playful – and create intrinsically motivating, meaningful environments which develop creative and resilient learners of the future.</p><img src="https://counter.theconversation.com/content/94850/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Nicola Whitton receives funding from the European Union. </span></em></p>We shouldn’t save play for the playground.Nicola Whitton, Professor in Education, Manchester Metropolitan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/855832017-10-17T00:44:45Z2017-10-17T00:44:45ZDo gamers behave the way game theory predicts they should?<figure><img src="https://images.theconversation.com/files/190017/original/file-20171012-31408-5ir9tn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How do people make complex decisions?</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/direction-decision-chance-opportunity-intersection-concept-261757220">Rawpixel.com/Shutterstock.com</a></span></figcaption></figure><p>When faced with a decision, people have varying ways of analyzing the choices. Give many people the same information, and they’ll all think about the situation differently, and often will choose slightly different options. As economists, we want to learn more about how people perceive and solve problems – including what sorts of situations are too difficult for people to analyze well. </p>
<p>Of particular interest is how people act in situations with <a href="http://www.artofstrategy.net/">two decision-makers whose choices influence each other</a>. These can be extremely common events like choosing how to dress on a first date, commercial options like setting prices or trade policies or even choices with global repercussions such as threatening to use nuclear weapons.</p>
<p>Our research has involved asking people to play a smartphone game presenting simplified versions of these situations; they have fun, and we gather data about how they make decisions. What we learn will help us and other researchers develop better theories of human behavior and problem-solving, which can help people make better decisions, governments design better policies and businesses make better managerial choices.</p>
<h2>Simulating choices</h2>
<p>The game, which we call “<a href="http://www.bluesandreds.com/">Blues and Reds</a>,” presents a more basic example of a potentially complex situation. Take, for instance, the potential for war between the <a href="http://www.newsweek.com/topic/u.s.-north-korea-relations">U.S. and North Korea</a>. One way to analyze the countries’ interactions is through the <a href="https://mitpress.mit.edu/books/course-game-theory">economic field of game theory</a> – in which a situation is presented as a series of interrelated choices.</p>
<p>A simple simulation might imagine, for instance, that North Korea has two potential targets, Guam and Hawaii, but only enough military power to attack one. And the scenario might be constructed so that a successful attack on just one of them would deliver a devastating defeat for the U.S. and an incredible victory for North Korea. But the U.S. gets to choose how to defend its territory. In this simplification, economists would imagine there are three choices for the U.S.: send all its defense to Guam, send everything to Hawaii or split defenses half and half between the two.</p>
<p>Game theory suggests creating a diagram called a <a href="https://doi.org/10.1016/S1574-0005(05)80005-0">tree</a>, indicating all the choices and the potential outcomes.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=426&fit=crop&dpr=1 600w, https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=426&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=426&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=535&fit=crop&dpr=1 754w, https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=535&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/190040/original/file-20171012-31381-1igu4us.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=535&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A decision tree diagram for a hypothetical, simplified conflict between the U.S. and North Korea.</span>
<span class="attribution"><span class="source">The Conversation, via LucidChart</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>This type of game theory analysis provides the opportunity to use a concept called “<a href="https://doi.org/10.1016/S0899-8256(05)80015-6">backward induction</a>” – looking to future options to guide choices in the present. In this case, the U.S. makes its defense decision, and then North Korea chooses where to attack. A quick look at the chart indicates clearly that if the U.S. defends only one location, North Korea will attack the other, and the U.S. will lose. The only way the U.S. can win in this simulation is if it defends both places.</p>
<p>However, like U.S.-North Korean relations, real-life interactive scenarios are much more complex, with more rounds of choices, more options and more players. </p>
<h2>A look at our game</h2>
<p>Our game “Blues and Reds” doesn’t present players with problems carrying that much geopolitical significance. But it does allow us to study how people actually behave when using trees and backward induction. Research tells us that <a href="http://www.sciencedirect.com/science/journal/00220531/104/1?sdc=1">people struggle when trees become too large</a>. (It’s hard to blame them: The decision tree for the relatively basic game of chess is so big that if every potential option were marked by a single atom, there are <a href="http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf">not enough atoms in the observable universe</a> to complete it.)</p>
<p>“Blues and Reds” consists of 58 puzzles against the computer. In each puzzle, the player moves a ball, which we call a “RoboToken,” one step through a puzzle, and then the computer responds. The player wins if the game ends with the RoboToken on a blue circle as opposed to a red one. In more complex levels, player and computer have multiple rounds of choices to make.</p>
<p>The game presents two different puzzles, which our players have nicknamed “snowflakes” and “rails.”</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/189840/original/file-20171011-28000-z6jqux.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A game version of a snowflake-like decision tree.</span>
<span class="attribution"><a class="source" href="http://www.bluesandreds.com/">Konrad Grabiszewski</a>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>“Snowflake” puzzles show the trees themselves, and let the player and the computer take turns moving the RoboToken from one step in the tree to the next. Players who win these puzzles do so by using backward induction, looking at possible outcomes and making choices that maximize their chances of getting to a blue circle. Tracking how many players succeed lets us know how capable they are of making good decisions when information is presented in this way. </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/189841/original/file-20171011-27991-1j1edcx.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A more complex depiction of an interaction between two players.</span>
<span class="attribution"><a class="source" href="http://www.bluesandreds.com/">Konrad Grabiszewski</a>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>“Rails” puzzles, on the other hand, are the same sorts of situations but depicted in a more complex way. As in the case of snowflakes, this is a turn-based puzzle with the player aiming to land the RoboToken on a blue node. As people work on those puzzles, we can evaluate whether they actually perceive these problems as trees.</p>
<h2>Why mobile experiments?</h2>
<p>Constructing a research experiment as a game makes pretty obvious sense when studying game theory. But making it a mobile app also dramatically improves our research quality. The population being studied can be much larger than standard study pools: Another mobile research game app, “<a href="http://www.seaheroquest.com/en">Sea Hero Quest</a>,” which helps with dementia studies, has been used by more than <a href="https://www.engadget.com/2017/08/30/sea-hero-quest-vr/">3 million</a> players. And people who want to participate in the research don’t have to live near where we work or take time out of their busy lives. They can just download a game to play in their spare time. </p>
<p>Our study can include anyone in the world who has a smartphone. Already, we have more than 10,000 users from more than 100 countries. We’ve only begun to analyze the data they’ve provided, but we expect to learn which puzzles are more difficult to solve (as measured by the percentage of users who win a puzzle) and, more importantly, what mathematical aspects make them easier or harder. For instance, we suspect that snowflake puzzles are easier than the rails, but we want to learn why. </p>
<p>As we refine our research, we can update the game to deepen our findings. We’re just at the beginning of finding out what we can learn from this game. Together, we and our game players will help decide the best ways to use mobile technology for social science research.</p><img src="https://counter.theconversation.com/content/85583/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Watching how people play a game against a computer opponent can help identify how humans use – or don’t use – game theory principles to make decisions.Konrad Grabiszewski, Visiting Assistant Professor of Economics, University of MiamiAlex Horenstein, Assistant Professor of Economics, University of MiamiLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/802562017-08-02T01:20:25Z2017-08-02T01:20:25ZThis math puzzle will help you plan your next party<figure><img src="https://images.theconversation.com/files/179645/original/file-20170725-30152-1sg6hk3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Mapping connections at your next shindig.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/unclibraries_commons/23109914111/in/photolist-Bd9mk2-4id8ef-9b3RB-7qFUhu-8ETfQv-kUVHxH-5ENkBs-4aWEEf-kUVF2v-kYPf5X-kYPdvK-8Y91FT-pKLRm-4aSCNx-bt2FqY-kYQmxU-5NXrDW-5QKVRi-668pD-iDChA-bS7NkP-7Qm21B-axnNjC-9fw57s-4jAkj1-kYQAZQ-aigLLX-8nwh4d-wrhJb-kYPqdv-6YW7NP-aigLEx-6Z199Q-ixZBQg-HyAgcu-kYPqZR-9v1r5W-ddzLaV-derxa6-5eiWhs-9cD5ss-auQjZo-cBiqX-9X9PGL-3JChmw-89RRvG-51s3TR-9P7eTG-7jhz8Z-4aSD1t/">unclibraries_commons</a></span></figcaption></figure><p>Let’s say you’re planning your next party and agonizing over the guest list. To whom should you send invitations? What combination of friends and strangers is the right mix? </p>
<p>It turns out mathematicians have been working on a version of this problem for nearly a century. Depending on what you want, the answer can be complicated. </p>
<p>Our book, <a href="http://press.princeton.edu/titles/10314.html">“The Fascinating World of Graph Theory</a>,” explores puzzles like these and shows how they can be solved through graphs. A question like this one might seem small, but it’s a beautiful demonstration of how graphs can be used to solve mathematical problems in such diverse fields as the sciences, communication and society.</p>
<h1>A puzzle is born</h1>
<p>While it’s well-known that Harvard is one of the top academic universities in the country, you might be surprised to learn that there was a time when Harvard had one of the nation’s best football teams. But in 1931, led by <a href="http://www.thecrimson.com/article/1971/3/12/barry-wood-31-was-star-for/">All–American quarterback Barry Wood</a>, such was the case. </p>
<p>That season Harvard played Army. At halftime, unexpectedly, Army led Harvard 13–0. Clearly upset, Harvard’s president told Army’s commandant of cadets that while Army may be better than Harvard in football, Harvard was superior in a more scholarly competition.</p>
<p>Though Harvard came back to defeat Army 14-13, the commandant accepted the challenge to compete against Harvard in something more scholarly. It was agreed that the two would compete – in mathematics. This led to Army and Harvard selecting mathematics teams; the showdown occurred in West Point in 1933. To Harvard’s surprise, Army won. </p>
<p>The Harvard–Army competition eventually led to an annual mathematics competition for undergraduates in 1938, called the <a href="https://www.maa.org/programs/maa-awards/putnam-competition-individual-and-team-winners">Putnam exam</a>, named for William Lowell Putnam, a relative of Harvard’s president. This exam was designed to stimulate a healthy rivalry in mathematics in the United States and Canada. Over the years and continuing to this day, this exam has contained many interesting and often challenging problems – including the one we describe above.</p>
<h1>Red and blue lines</h1>
<p>The 1953 exam contained the following problem (reworded a bit): There are six points in the plane. Every point is connected to every other point by a line that’s either blue or red. Show that there are three of these points between which only lines of the same color are drawn. </p>
<p>In math, if there is a collection of points with lines drawn between some pairs of points, that structure is called a graph. The study of these graphs is called graph theory. In graph theory, however, the points are called vertices and the lines are called edges.</p>
<p>Graphs can be used to represent a wide variety of situations. For example, in this Putnam problem, a point can represent a person, a red line can mean the people are friends and a blue line means that they are strangers.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=266&fit=crop&dpr=1 600w, https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=266&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=266&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=335&fit=crop&dpr=1 754w, https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=335&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/179639/original/file-20170725-30149-in14d6.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=335&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Show that there are three points connected by lines of the same color.</span>
<span class="attribution"><span class="source">Gary Chartrand</span></span>
</figcaption>
</figure>
<p>For example, let’s call the points A, B, C, D, E, F and select one of them, say A. Of the five lines drawn from A to the other five points, there must be three lines of the same color. </p>
<p>Say the lines from A to B, C, D are all red. If a line between any two of B, C, D is red, then there are three points with only red lines between them. If no line between any two of B, C, D is red, then they are all blue.</p>
<p>What if there were only five points? There may not be three points where all lines between them are colored the same. For example, the lines A–B, B–C, C–D, D–E, E–A may be red, with the others blue.</p>
<p>From what we saw, then, the smallest number of people who can be invited to a party (where every two people are either friends or strangers) such that there are three mutual friends or three mutual strangers is six. </p>
<p>What if we would like four people to be mutual friends or mutual strangers? What is the smallest number of people we must invite to a party to be certain of this? This question has been answered. It’s 18. </p>
<p>What if we would like five people to be mutual friends or mutual strangers? In this situation, the smallest number of people to invite to a party to be guaranteed of this is – unknown. Nobody knows. While this problem is easy to describe and perhaps sounds rather simple, it is notoriously difficult.</p>
<h1>Ramsey numbers</h1>
<p>What we have been discussing is a type of number in graph theory called a Ramsey number. These numbers are named for the British philosopher, economist and mathematician <a href="https://www.repository.cam.ac.uk/bitstream/handle/1810/3484/RamseyText.html?sequence=5">Frank Plumpton Ramsey</a>. </p>
<p>Ramsey died at the age of 26 but obtained at his very early age a very curious theorem in mathematics, which gave rise to our question here. Say we have another plane full of points connected by red and blue lines. We pick two positive integers, named r and s. We want to have exactly r points where all lines between them are red or s points where all lines between them are blue. What’s the smallest number of points we can do this with? That’s called a Ramsey number. </p>
<p>For example, say we want our plane to have at least three points connected by all red lines and three points connected by all blue lines. The Ramsey number – the smallest number of points we need to make this happen – is six. </p>
<p>When mathematicians look at a problem, they often ask themselves: Does this suggest another question? This is what has happened with Ramsey numbers – and party problems. </p>
<p>For example, here’s one: Five girls are planning a party. They have decided to invite some boys to the party, whether they know the boys or not. How many boys do they need to invite to be certain that there will always be three boys among them such that three of the five girls are either friends with all three boys or are not acquainted with all three boys? It’s probably not easy to make a good guess at the answer. It’s 41!</p>
<p>Very few Ramsey numbers are known. However, this doesn’t stop mathematicians from trying to solve such problems. Often, failing to solve one problem can lead to an even more interesting problem. Such is the life of a mathematician.</p><img src="https://counter.theconversation.com/content/80256/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Let’s say you want the perfect mix of friends and strangers at your next party. Mathematicians have been working on a version of this problem for nearly a century, and the answer is complicated.Gary Chartrand, Professor Emeritus of Mathematics, Western Michigan UniversityArthur Benjamin, Professor of Mathematics, Harvey Mudd CollegePing Zhang, Professor of Mathematics, Western Michigan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/812882017-07-20T11:07:07Z2017-07-20T11:07:07ZLions and lambs: can you solve this classic game theory puzzle?<figure><img src="https://images.theconversation.com/files/178995/original/file-20170720-23992-o8glv9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>How many lions does it take to kill a lamb? The answer isn’t as straightforward as you might think. Not, at least, according to game theory.</p>
<p><a href="https://theconversation.com/economic-theories-that-have-changed-us-game-theory-43633">Game theory</a> is a branch of maths that studies and predicts decision-making. It often involves creating hypothetical scenarios, or “games”, whereby a number of individuals called “players” or “agents” can choose from a defined set of actions according to a series of rules. Each action will have a “pay-off” and the aim is usually to find the maximum pay-off for each player in order to work out how they would likely behave.</p>
<p>This method has been used in a wide variety of subjects, including <a href="http://www.economicsonline.co.uk/Business_economics/Oligopoly.html">economics</a>, <a href="https://www.nature.com/scitable/knowledge/library/game-theory-evolutionary-stable-strategies-and-the-25953132">biology</a>, <a href="http://www.jstor.org/stable/139591">politics</a> and <a href="https://en.wikipedia.org/wiki/Behavioral_game_theory">psychology</a>, and to help explain behaviour in auctions, voting and market competition. But game theory, thanks to its nature, has also given rise to some entertaining brain teasers.</p>
<p>One of the less famous of these puzzles involves working out how players will compete over resources, in this case hungry lions and a tasty lamb. A group of lions live on an island covered in grass but with no other animals. The lions are identical, perfectly rational and aware that all the others are rational. They are also aware that all the other lions are aware that all the others are rational, and so on. This mutual awareness is what’s referred to as “<a href="https://plato.stanford.edu/entries/common-knowledge/">common knowledge</a>”. It makes sure that no lion would take a chance or try to outsmart the others.</p>
<p>Naturally, the lions are extremely hungry but they do not attempt to fight each other because they are identical in physical strength and so would inevitably all end up dead. As they are all perfectly rational, each lion prefers a hungry life to a certain death. With no alternative, they can survive by eating an essentially unlimited supply of grass, but they would all prefer to consume something meatier.</p>
<p>One day, a lamb miraculously appears on the island. What an unfortunate creature it seems. Yet it actually has a chance of surviving this hell, depending on the number of lions (represented by the letter N). If any lion consumes the defenceless lamb, it will become too full to defend himself from the other lions.</p>
<p>Assuming that the lions cannot share, the challenge is to work out whether or not the lamb will survive depending on the value of N. Or, to put it another way, what is the best course of action for each lion – to eat the lamb or not eat the lamb – depending on how many others there are in the group.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=502&fit=crop&dpr=1 754w, https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=502&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/178996/original/file-20170720-24011-8n7826.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=502&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">N=3.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<h2>The solution</h2>
<p>This type of game theory problem, where you need to find a solution for a general value of N (where N is a positive whole number), is a good way of testing game theorists’ logic and of demonstrating how backward induction works. Logical induction involves using evidence to form a conclusion that is probably true. <a href="http://www.gametheory.net/dictionary/BackwardInduction.html">Backward induction</a> is a way of finding a well-defined answer to a problem by going back, step-by-step, to the very basic case, which can be solved by a simple logical argument. </p>
<p>In the lions game, the basic case would be N=1. If there was only one hungry lion on the island it would not hesitate to eat the lamb, since there are no other lions to compete with it.</p>
<p>Now let’s see what happens in the case of N=2. Both lions conclude that if one of them eats the lamb and becomes too full to defend itself, it would be eaten by the other lion. As a result, neither of the two would attempt to eat the lamb and all three animals would live happily together eating grass on the island (if living a life solely dependent on the rationality of two hungry lions can be called happy).</p>
<p>For N=3, if any one of the lions eats the lamb (effectively becoming a defenceless lamb itself), it would reduce the game to the same scenario as for N=2, in which neither of the remaining lions will attempt to consume the newly defenceless lion. So the lion that is closest to the actual lamb, eats it and three lions remain on the island without attempting to murder each other.</p>
<p>And for N=4, if any of the lions eat the lamb, it would reduce the game to the N=3 scenario, which would mean that the lion that ate the lamb would end up being eaten itself. As none of the lions want that to happen, they leave the lamb alone.</p>
<p>Essentially, the outcome of the game is decided by the action of the lion closest to the lamb. For each integer N, the lion realises that eating the lamb would reduce the game to the case of N-1. If the N-1 case results in the survival of the lamb, the closest lion eats it. Otherwise, all the lions let the lamb live. So, following the logic back to the base case every time, we can conclude that the lamb will always be eaten when N is an odd number and will survive when N is an even number.</p><img src="https://counter.theconversation.com/content/81288/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Amirlan Seksenbayev does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Put a lamb on an island of lions and they’ll eat it – or will they?Amirlan Seksenbayev, PhD Candidate in Mathematical Sciences, Probability and Applications, Queen Mary University of LondonLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/753032017-06-18T19:52:30Z2017-06-18T19:52:30ZYour guide to solving the next online viral maths problem<figure><img src="https://images.theconversation.com/files/174130/original/file-20170616-565-1v5w26n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">You need to think like a mathematician to solve those viral maths problems.</span> <span class="attribution"><span class="source">Shutterstock/Dean Drobot</span></span></figcaption></figure><p>How many times have you seen a post online or part of your social media feed that says something like “<a href="https://mic.com/articles/143062/this-math-problem-is-stumping-the-whole-internet-can-you-solve-it#.zfJUBEV19">This Math Problem Is Stumping the Whole Internet. Can You Solve It?</a>” or “<a href="https://www.facebook.com/tara.haelle/posts/568740469816867">Apparently 9 out of 10 people get this wrong. Do you know the answer?</a>”</p>
<p>At the heart of the post is usually a problem involving numbers and symbols, such as this one:</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"631405920078073856"}"></div></p>
<p>They’re usually followed by plenty of <a href="https://www.reddit.com/r/AskReddit/comments/1b9n6u/i_am_really_confused_why_do_some_people_say_the/">discussion online</a> about the <a href="https://community.spiceworks.com/topic/137542-what-is-the-answer-to-the-equation-6-2-1-2-x-is-it-9-or-1">various possible answers</a> to the problem, including questions about why some people think even <a href="https://productforums.google.com/forum/#!topic/websearch/kZkTv_WTSxA">Google’s online calculator gets the wrong answer</a>.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"499506580258168832"}"></div></p>
<p>This is just one of many examples of similar problems that have been doing the rounds for years but <a href="https://www.reddit.com/r/AskReddit/comments/6gb0hb/what_the_heck_is_the_answer_to_6221/">still continue to baffle some people</a>. Here’s another:</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"770686417857024000"}"></div></p>
<p>No matter how hard you try, it’s impossible to resist that challenge. You give it a go and then look at the comments section only to find some people agree with your answer while others have something completely different. </p>
<p>So let me outline the correct way to approach these online equations with the minimum of fuss. I’ll explain why in some cases there may be more than one possible correct answer.</p>
<h2>The language of mathematics</h2>
<p>In the English language we read from left to right. It therefore seems very natural to look at mathematical equations in the same way. </p>
<p>But you wouldn’t try to read Mandarin or Arabic like this, and nor should you attempt to do so with the distinct language of mathematics. </p>
<p>To be maths-literate, it is important to understand the relevant rules about “spelling” and “grammar” in mathematics.</p>
<p>A strict set of rules known as the <a href="https://www.mathsisfun.com/operation-order-bodmas.html">order of operations</a> defines the correct arithmetical grammar. These rules tell us the order in which we must perform mathematical operations such as addition and multiplication when both appear in an equation.</p>
<p>In Australia, the mnemonic BODMAS (Brackets, Order, Division, Multiplication, Addition, Subtraction) is typically taught to students to help them remember the correct order. Here, the ‘Order’ in BODMAS refers to mathematical powers such as squared, cubed or square root. </p>
<p>In other countries, this may be taught as <a href="http://www.mathsisfun.com/operation-order-pemdas.html">PEMDAS</a>, <a href="http://www.mathsisfun.com/operation-order-bodmas.html">BEDMAS or BIDMAS</a>, but these all boil down to exactly the same thing.</p>
<p>This means that if, for example, we have an equation that contains both addition and multiplication, we always carry out multiplication first regardless of the order in which they are written.</p>
<p>Consider the following equations:</p>
<blockquote>
<p>(a) 3×4+2</p>
<p>(b) 2+3×4 </p>
</blockquote>
<p>When we apply BODMAS, we can see that these equations are exactly the same (or equivalent) – in both cases we begin by calculating 3×4=12, then compute 12+2=14.</p>
<p>But some people are likely to get the wrong answer for the second equation because they will try to solve it from left to right. They will do the addition first (2+3=5) and then multiplication (5×4) to obtain an incorrect answer of 20.</p>
<h2>Brackets can make a difference</h2>
<p>This is where brackets (or parentheses) can be a very useful part of arithmetical punctuation. In English, a well-placed comma can be the difference between saying “Let’s eat, John” and “Let’s eat John”.</p>
<p>The same applies in maths, where a well-placed bracket can completely change our calculation. Brackets are used to give priority to a particular part of an equation – we always carry out the calculation inside the bracket before dealing with what is outside.</p>
<p>If we introduce brackets around the addition in equations (a) and (b) above, then we have two new equations:</p>
<blockquote>
<p>(c) 3×(4+2)</p>
<p>(d) (2+3)×4</p>
</blockquote>
<p>These equations are no longer equivalent to each other. In both cases, the brackets tell us to do the addition before we do the multiplication. This means we have to calculate 3×6 for (c) and 5×4 for (d). We now get different answers, (c) is 18 and (d) is 20. </p>
<p>Note that for equations (a) and (b), brackets were not necessary because BODMAS tells us to carry out multiplication before addition anyway. However, adding brackets that reinforce the BODMAS rules can help to avoid any confusion.</p>
<h2>More rules</h2>
<p>Understanding BODMAS gets us most of the way there in terms of solving these problems, but it also helps to be aware of the commutative and associative properties of mathematics.</p>
<p>A mathematical operation is commutative if it does not matter which order the operands (numbers) are written in. Addition is commutative, since a+b=b+a.</p>
<p>But subtraction is not, because a-b is not the same as b-a. It is also straightforward to show that multiplication is commutative, but division is not. </p>
<p>Such distinctions exist in the English language too. Ordering “vodka and orange juice” is the same as ordering “orange juice and vodka”, but “shaken not stirred” is not the same as “stirred not shaken”.</p>
<p>An operation is associative if, when we have multiple consecutive occurrences of this operation, it does not matter which order we carry them out in. </p>
<p>Again, addition and multiplication have this property, while subtraction and division do not. If we have the equation a+b+c, then it does not matter whether we solve it as (a+b)+c or a+(b+c).</p>
<p>But if we have a-b-c then the order is important, as (a-b)-c is not the same as a-(b-c) and we should always work from left to right. See for youself:</p>
<blockquote>
<p> (3-2)-1=0</p>
<p> 3-(2-1)=2</p>
</blockquote>
<p>Again, English language implicitly has such concepts; “rum and coke and lime” is the same product regardless of whether rum is added to (coke and lime), or lime is added to a (rum and coke).</p>
<p>But we cannot rearrange any of these operations in “order then drink then leave” – a successful trip to the pub relies on these actions being carried out in exactly that order.</p>
<p>Once we understand the correct order of operations and the associative and commutative properties, we have the toolbox to solve any simple, well-defined arithmetical equation.</p>
<h2>So do you know the answer?</h2>
<p>So let’s return to the original problem:</p>
<blockquote>
<p> 6÷2(1+2)=?</p>
</blockquote>
<p>The equation has more than one legitimate meaning. Some might believe the answer is 1, others might think the answer is 9. And neither answer is really wrong.</p>
<p>After carrying out the addition inside the brackets, we are left with 6÷2(3). Some people will argue that we should work from left to right, calculating 6÷2=3 and then multiplying 3×3=9, which is the answer given by Google’s calculator.</p>
<p>Others, and I consider myself part of this camp, would argue that 2(1+2) should be computed in its entirety first, since the juxtaposition of these terms without a × sign implies that it consists of a single element.</p>
<p>A mathematician would more normally express the equation as follows:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=100&fit=crop&dpr=1 600w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=100&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=100&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=126&fit=crop&dpr=1 754w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=126&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=126&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>That leaves us with 6÷6=1.</p>
<p>The problem here is a poorly constructed equation, and the ÷ is the main culprit. Mathematicians rarely use this sign (or the multiplication sign ×); in practice, they prefer to use clear, unambiguous notation such as fractions.</p>
<p>If we want to convey the first meaning above, it would be more common to write the equation with extra brackets as (6/2)(1+2), which gives the answer 9. To convey the second meaning, we would write 6/(2(1+2)) as shown in the equation above, which gives the answer 1.</p>
<p>By writing things this way, we can eliminate the whole debate and save everyone a lot of time and energy.</p>
<p>Online puzzles can be a great way to refresh your mathematical skills, but it’s important to watch out for the deliberately confusing ones. </p>
<p>The next time one pops up on your timeline, remember BODMAS and you should be fine. </p>
<p>But if the answer is still not clear, then it’s best to avoid the debate and instead step back, take a deep breath and say: “They haven’t spelled that correctly!”</p><img src="https://counter.theconversation.com/content/75303/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Craig Anderson does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>There’s a reason why some people get different answers to those frustrating viral maths problems. You need to learn how to “read” the maths.Craig Anderson, Postdoctoral Research Fellow in Statistics, University of Technology SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/775562017-05-12T03:56:06Z2017-05-12T03:56:06ZRemembering Bill Tutte: another brilliant codebreaker from World War II<figure><img src="https://images.theconversation.com/files/169044/original/file-20170511-32578-rxuetz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bill Tutte, the brilliant codebreaker.</span> <span class="attribution"><span class="source">Newmarket Journal </span></span></figcaption></figure><p>One of the greatest mathematicians and codebreakers of the 20th century, William (Bill) Tutte, was born a century ago this Sunday, May 14.</p>
<p>His wartime work enabled the British to break into the communications of the highest levels of the Nazi regime, motivated the development of a special-purpose electronic codebreaking computer, shortened World War II and saved countless lives.</p>
<p>Tutte, <a href="https://www.theguardian.com/news/2002/may/10/guardianobituaries.obituaries">who died aged 84 in Canada in 2002</a>, went on to do far-reaching work in mathematics but few people have heard of him and his contributions.</p>
<h2>First breakthrough</h2>
<p>Tutte’s origins were humble. He was born in Newmarket, a market town in England north of London, the son of a gardener and a housekeeper.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=414&fit=crop&dpr=1 600w, https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=414&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=414&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=520&fit=crop&dpr=1 754w, https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=520&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/169045/original/file-20170511-32596-pf440b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=520&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The young Bill Tutte (bottom row, right) at Cheveley Village School.</span>
<span class="attribution"><span class="source">Newmarket Journal</span></span>
</figcaption>
</figure>
<p>He excelled at school and entered Trinity College, Cambridge, in 1935, where he majored in chemistry.</p>
<p>While still an undergraduate he became close friends with three mathematics students: Leonard Brooks, Cedric Smith and Arthur Stone. Together, these four threw themselves into mathematical problem-solving and research.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168860/original/file-20170511-21603-endjc0.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Squaring the square: the lowest-order perfect squared square.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Squaring_the_square.svg">Wikimedia</a></span>
</figcaption>
</figure>
<p>They were attracted to a simple recreational puzzle, on whether it is possible to divide a square up into smaller squares, all of different sizes, known as <a href="http://www.squaring.net/history_theory/history_theory.html">Squaring the Square</a>. </p>
<p>The prevailing belief was that it could not be done. But they managed to do it, partly by discovering an unexpected link with the mathematics of electrical circuits.</p>
<p>The theoretical framework they developed has had a lasting influence. A German mathematician, Roland Sprague, working independently, just pipped them to a solution to the puzzle, but not the theory behind it.</p>
<p>The work of Tutte and his friends was published in an academic journal in 1940. It got Tutte noticed at Cambridge, and from there he joined Britain’s wartime codebreaking operation at <a href="https://www.bletchleypark.org.uk/">Bletchley Park</a> in 1941.</p>
<h2>The codebreakers</h2>
<p>Other Cambridge mathematicians were there before Tutte. Among them was <a href="https://theconversation.com/the-imitation-game-is-it-history-drama-or-myth-35849">Alan Turing</a>, who had worked out how to break the version of the <a href="https://www.bletchleypark.org.uk/our-story/the-challenge/enigma">Enigma</a> code used by the German navy.</p>
<p>The Enigma code was already so difficult that even there, at Bletchley Park – the best codebreaking operation of the War – it had sat in the too-hard basket until Turing’s arrival. It was a very tough problem, even for him.</p>
<p>Tutte worked on different cypher machine, known as the <a href="https://www.bletchleypark.org.uk/our-story/the-challenge/lorenz">Lorenz</a> cypher. This was the one used by the Nazi High Command, including Hitler himself.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/169050/original/file-20170512-32624-eimvv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The Nazis’ Lorenz machine.</span>
<span class="attribution"><span class="source">Bletchley Park/Shaun Armstrong</span></span>
</figcaption>
</figure>
<p>It was much more complex than Enigma, and on top of that, the British knew very little about how it worked, whereas with Enigma they knew everything. </p>
<p>So it was a harder problem with less information, and yet Tutte solved it. It was a staggering achievement.</p>
<p>Tutte’s breakthrough was based on careful analysis of intercepted encrypted traffic to identify some periodic behaviour that indicated the size of a “wheel” component in the machine.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/bMu8UiHJHgs?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Tutte’s attack on Lorenz needed to be automated. This led to the design and construction of the <a href="http://www.tnmoc.org/explore/colossus-gallery">Colossus</a> machines, led by <a href="http://www.independent.co.uk/arts-entertainment/obituary-tommy-flowers-1184727.html">Tommy Flowers</a>. </p>
<p>These are sometimes regarded as the world’s first computers, having most of the fundamental characteristics that the term “computer” is taken to embrace today.</p>
<p>They were so successful at breaking into the encoded messages of the Nazi regime’s high command that they were often able to decode the messages at the same time as the intended German recipients were reading them. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/169048/original/file-20170512-32578-12x2a3q.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The Lorenz machine on show to visitors at Bletchley Park.</span>
<span class="attribution"><span class="source">Bletchley Park/Shaun Armstrong</span></span>
</figcaption>
</figure>
<p>This gave an incalculable advantage to the Allies in the later years of the war, including their preparations for invading Normandy.</p>
<p>Tutte’s work on Lorenz has been described as the greatest intellectual achievement of the second world war. As a result, he was given a fellowship at Cambridge and went on to do his PhD there.</p>
<h2>From codes to networks</h2>
<p>Another big effect of the Squaring the Square puzzle was to turn Tutte’s attention more to mathematics, while he was majoring in chemistry, and in particular to the theory of graphs. </p>
<p>These are not the simple graphs you would use to chart things such as daily temperatures over time. Rather, they are abstract networks, consisting of objects (called vertices or nodes), and interactions between them (called edges or links).</p>
<p>Think of the network of train stations, together with the rail lines between them. Or we might have web pages, with hyperlinks between them, making the graph we know as the World Wide Web. We might have people, with friendships between them. And so on.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=398&fit=crop&dpr=1 600w, https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=398&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=398&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=501&fit=crop&dpr=1 754w, https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=501&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/169054/original/file-20170512-32613-1vgymvj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=501&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Bill Tutte in the 1960s.</span>
<span class="attribution"><span class="source">Richard Youlden</span></span>
</figcaption>
</figure>
<p>One of Tutte’s major contributions was to determine the exact place of graphs among some other mathematical objects for which more theory was known.</p>
<p>In mathematics, the simplest objects to deal with are those that are straight and flat – for example, lines and planes. We call such things linear.</p>
<p>Much of mathematics is about taking things that are non-linear – meaning that they are curved or bent rather than straight or flat – and trying to make them linear, or nearly linear, or to replace them with something linear. </p>
<p>For example, mathematicians can study curves by zooming in so that they look straight, or close to it.</p>
<p>Now, graphs – or networks – are much more complex objects than simple straight lines or planes. Nevertheless, it turns out that there are linear ways of looking at them.</p>
<h2>Extra dimensions</h2>
<p>But these come at a price. You have to work in many many dimensions, not just the three dimensions of space that we are used to. It’s hard for us to imagine these extra dimensions, as they represent directions that are so weird, so outside our universe, that we can’t even point in them.</p>
<p>In these vast multidimensional worlds, it’s not so easy to tell graphs apart from other linear objects you find. This is what Tutte showed us how to do. He pinpointed exactly what was special about graphs. </p>
<p>This theory brought a new depth to the subject, and related this new field to older and more developed parts of mathematics.</p>
<p>So it was that a purely recreational problem, a source of mathematical fun for undergraduates, sowed seeds that grew into a major contribution to overthrowing the Nazi regime, and raised up a new branch of mathematics that is now used to understand the complex networks that permeate the modern world.</p>
<p>More than a decade after Tutte’s death a <a href="http://www.newmarketjournal.co.uk/news/new-sculpture-honours-forgotten-war-hero-1-6290664">memorial was erected</a> in his former home town of Newmarket.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=689&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=689&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=689&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=866&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=866&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168862/original/file-20170511-21613-1wwkgo8.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=866&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Memorial to Bill Tutte in Newmarket.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/nickhubbard/32798700014/">Flickr/Nick Hubbard</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<hr>
<p><em>This article is based on a <a href="https://thelaborastory.com/stories/william-thomas-tutte/">talk by the author at The LaboraStory</a>. For information on any Tutte Centenary events, including an event at Bletchley Park on Sunday May 14, see <a href="http://billtuttememorial.org.uk/centenary/">the Bill Tutte Memorial Fund</a>.</em></p><img src="https://counter.theconversation.com/content/77556/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Graham Farr has received funding from the Australian Research Council.</span></em></p>He was one of the brilliant mathematical geniuses who helped crack the Nazi codes, but few have ever heard of his name. So who was Bill Tutte?Graham Farr, Professor, Faculty of IT, Monash UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/645082016-08-31T13:26:05Z2016-08-31T13:26:05ZStudy reveals what it takes to become a cryptic crossword expert – and it’s more than just practice<figure><img src="https://images.theconversation.com/files/136081/original/image-20160831-30801-18px47x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/chrisblanar/2163230762/in/photolist-4ia897-8gHgVg-hcAVeD-34Eg3Y-2obBPr-34zGtP-5WNbx6-5GhQe7-8tzt7u-9yPgLe-5b8ASb-dKVabi-5b4ivX-5X2r3k-ordhu2-bbp3o-bWiT6-7EJqmT-aAwbeJ-5qjc8o-q2Jx4h-brffix-5vJYPa-5qeRXk-9kzoaM-hAGJ5d-5qeRSr-5qjc6E-9rmE6K-9cdgCV-odqCPQ-96TkjH-24uyiw-ixzi7S-cXSnA-58BrW9-4MwLtM-9uKCf7-nPVpzx-6bJYsf-6bJZDw-EDMDL-pJHonp-5a9RXm-Dooeto-4SfiQN-cNzMVJ-33FESE-e8z486-qpuoQb">Chris Blanar/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span></figcaption></figure><p>You may have heard of the “10,000-hour rule”, the belief that it takes thousands of hours of intense practice to become an expert in something. Training and practice are clearly vitally important in many highly competitive areas such as <a href="https://hbr.org/2007/07/the-making-of-an-expert">sports, music and chess</a>. But is that really all it takes to achieve greatness? </p>
<p><a href="https://theconversation.com/theres-more-than-practice-to-becoming-a-world-class-expert-60430">Recent research</a> suggests that other factors such as genetics influence the likelihood that you will try, enjoy and excel at a performance activity. We decided to <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2016.00567/full">test that theory</a> in the highly challenging arena of cryptic crossword solving. What we’ve found so far suggests that these kinds of word puzzles actually attract people with an affinity for maths and science and that the ability to think flexibly seems more important than hours of practice when it comes to solving them.</p>
<p>Unlike regular crosswords, which typically ask the solver to find a synonym for a word or phrase, cryptic crosswords use clues that are deliberately misleading. Solvers have to ignore this reading and look instead for a grammatical set of coded instructions to lead them to the correct answer. The problem lies in recognising and cracking the code, and the task of the crossword setter, like that of a magician, is to conceal the mechanism so subtly that the way to the answer is hard to find.</p>
<p>Have a look at these cryptic clues <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2016.00567/full">from our paper</a> and see if you can spot where you are being misled. The answers and explanations are at the foot of the article if you need some help.</p>
<ul>
<li><p>Active women iron some skirts and shirts (9)</p></li>
<li><p>Speciality of the Cornish side that’s perfect with new wingers (5,4)</p></li>
</ul>
<h2>Different kind of expertise</h2>
<p>Cryptic crosswords are different from other activities previously studied to explore what it takes to become an expert. For example, unlike chess, sport or music, there are very few monetary rewards or prizes on offer, and nothing by way of global prestige. For this reason, we believed that daily solving regimes would be relatively short and relaxed, with none of the deliberate, arduous and unenjoyable training burdens that <a href="http://www.tandfonline.com/doi/abs/10.1080/1359813980090106">research has suggested</a> are needed for high expertise. </p>
<p>At the same time, we knew that there was a wide ability range in tackling cryptics. For example, Times Crossword Championship winners such as Mark Goodliffe can solve three tournament puzzles <a href="https://www.theguardian.com/crosswords/crossword-blog/2014/oct/20/crossword-blog-watching-a-champion-solver-at-work">in just 22 minutes</a>. On the other hand, others can take well over an hour for just one puzzle, even after decades of solving. As solvers don’t do any explicit training, unlike musicians or chess players, there was an opportunity to study what other factors might have led to these performance differences.</p>
<p>We also felt that previous expertise studies had overlooked a vitally important aspect: whether participants had specific characteristics that could explain why they enjoyed the activity. So, our first move was to get to know the crossword-solving population by conducting a very broad survey advertised through online crossword community sites, such as those providing <a href="http://www.fifteensquared.net/">answers and explanations</a> to the previous day’s crossword. As well as collecting typical demographic information, we asked solvers about their education, career, hobbies, why they solved crosswords, and whether they felt a need to engage in intellectually stimulating activities in their spare time.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/136050/original/image-20160831-29099-xe3o81.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Brain teaser.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>The results of the survey were very revealing. Cryptic crossword solvers seem to be academic high-flyers. Over 80% of the 805 respondents, regardless of whether they were experts or hobbyists, had a university degree (but typically went to university at a time when only 10% of the population did so), and 12% had PhDs. They seemed also to have a drive to think, an itchy brain they need to scratch whether in their hobbies or in their challenging careers.</p>
<p>We also found that solvers tend to be qualified in scientific fields such as mathematics, computing, chemistry and medicine. What’s more, this trend increased significantly with expertise. Some of our participants were “super-solvers” who could crack exceptionally hard crosswords or finish a tough cryptic in less than 15 minutes. Nearly a third of these solvers worked in IT, compared to just a fifth of hobbyists. Similarly, super-solvers were much more likely to have studied maths at university. </p>
<p>Although you might be forgiven for thinking that crossword solving was all about having a good vocabulary, this link to skill in maths does make sense. We’ve suspected for some time that solving cryptics is much more to do with a logical, code-cracking approach to the clues than having good verbal skills. This is probably why MI6 famously decided to turn to cryptic crossword solvers as a <a href="http://www.telegraph.co.uk/history/world-war-two/11151478/Could-you-have-been-a-codebreaker-at-Bletchley-Park.html">useful source of code crackers</a> for Bletchley Park during World War II.</p>
<p>Finally, our hunch that practice levels would be comparatively low turned out to be the case. The time spent solving crosswords amounted to just six to seven hours a week (one or two crosswords a day) with only 20 minutes spent on other crossword-related activity (mostly to do with looking up the answers online). This didn’t vary among the expertise groups and is considerably less practice time than experts spend on other researched areas of expertise <a href="http://dx.doi.org/10.1016/j.intell.2013.04.001">such as music</a>.</p>
<h2>Aptitude as well as practice</h2>
<p>Our survey strongly suggests that having a leaning towards maths, science and code-cracking and a strong desire to engage your brain even in your leisure time are key qualities among cryptic crossword solvers. What we are not claiming is that you need to have gone to university to do cryptics. In fact, most of our participants had already started to solve in their mid-teens. But our research does suggest that there is a minimum threshold of flexible problem-solving ability for tackling cryptic crosswords, which is being reflected indirectly in the very high levels of university participation.</p>
<p>As with all skills, learning to solve cryptic crosswords requires engagement over a number of years to acquire a good knowledge of the rules and the common clue types. But both the experts and the hobbyists in our survey had been solving for over three decades on average, yet had achieved quite different levels of solving performance. This really does suggest that aptitudes, as well as practice, are important in this area. </p>
<p>Our next step will be to look at crossword solving under lab conditions. This will help us to establish exactly which cognitive aspects highlighted by the survey allow experts to outperform their hobbyist peers. Then we’ll know exactly what it takes to crack a cryptic.</p>
<h2>Answers and explanations for the clues:</h2>
<p><strong>Active women iron some skirts and shirts (9)</strong></p>
<ul>
<li><p>The definition is “Active women” = an obliquely phrased straight definition for FEMINISTS.</p></li>
<li><p>The wordplay comprises: FE (iron, chemical symbol) + MINIS (plural form of a type of skirt, hence the word “some”) + TS (plural of “T”, an abbreviation for “T-Shirt”).</p></li>
<li><p>The surface meaning is highly misleading; the interpretation of iron relies on a linguistic ambiguity (noun, not verb).</p></li>
</ul>
<p><strong>Speciality of the Cornish side that’s perfect with new wingers (5,4)</strong></p>
<ul>
<li><p>The definition is “Speciality of the Cornish” = CREAM TEAS.</p></li>
<li><p>The wordplay comprises: DREAM TEAM (“side that’s perfect”) with D and M replaced by new letters on either edge (“with new wingers”).</p></li>
</ul><img src="https://counter.theconversation.com/content/64508/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>It could be you.Kathryn Friedlander, Lecturer in Creative Performance and Expertise, University of BuckinghamPhilip A. Fine, Senior Lecturer in Psychology, University of BuckinghamLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/524012015-12-16T11:06:30Z2015-12-16T11:06:30ZHow to crack British intelligence service’s devilish Christmas puzzle<figure><img src="https://images.theconversation.com/files/106072/original/image-20151215-23202-jzr0rd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Easy-peasy?</span> <span class="attribution"><a class="source" href="http://www.gchq.gov.uk/press_and_media/news_and_features/Pages/Directors-Christmas-puzzle-2015.aspx">GCHQ</a></span></figcaption></figure><p>Calling all aspiring spooks. Robert Hannigan, director of Britain’s security and intelligence organisation <a href="http://www.gchq.gov.uk/Pages/homepage.aspx">GCHQ</a>, has included <a href="http://www.gchq.gov.uk/press_and_media/news_and_features/Pages/Directors-Christmas-puzzle-2015.aspx">a rather tantalising puzzle</a> with his Christmas card this year. He hopes that it will exercise your grey cells over the holiday period.</p>
<p>If you can solve the puzzle, along with the others that it will lead to, you can email the solution to GCHQ (the Government communications headquarters) before January 31. A winner will be drawn from all the correct answers – and doubtless be named to much fanfare.</p>
<p>So what do you need to do to be in with a chance?</p>
<p>The puzzle requires that you shade in squares on the 25x25 grid shown above. But which ones? Well, a few of the “black” squares have been completed for you, but most you will have to do yourself. By way of a clue, each row and cell has a sequence of numbers attached to it. The numbers represent a sequence of shaded cells, that need to separated from each other by at least one blank cell. For example, the row marked “7 3 1 1 7” should contain a sequence of seven shaded cells, followed by at least one blank cell, then three shaded cells, followed by at least one blank cell – and so on. The problem is made trickier because each horizontal row intersects a vertical column, each with its own sequence code.</p>
<h2>Paper and pen</h2>
<p>So how do you reach the solution? One way of cracking it is to resort to old-fashioned paper and pen. Just sit down, put on your thinking cap and try to reason it out.</p>
<p>It is not that difficult to get started. In fact, it is already started, and it is easy to fill in a few more squares. Take a look at row 22 on the horizontal axis – the one that has the sequence “1 3 1 3 10 2”. These numbers add up to 20 and as there are six blocks, you need at least five blank squares to separate them. As we only have 25 squares in the row, this pattern can only fit in one way – the first square in the row has to be shaded, and the rest just follow, with only one blank square between each run of shaded squares.</p>
<p>Are there any others like this? Column seven (“7 1 1 1 1 1 7”) sums up to 19. As we have seven numbers we need at six least separators. This also adds up to 25, so this row is easy to complete, too. The figure below shows the grid once we have filled in row 22 and column seven (the blank squares are marked in yellow).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/106086/original/image-20151215-23176-1lga4db.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">State of grid after completing row 22 and column seven.</span>
</figcaption>
</figure>
<p>Any others where the row or column is also easy to complete? Yes, but I’ll leave it to you to find them.</p>
<p>Once you have completed the “easy” rows/columns, you then need to start looking for other ways of completing the remaining rows and columns, or even reasoning about individual cells. In many ways, it is like completing <a href="https://theconversation.com/good-at-sudoku-heres-some-youll-never-complete-5234">a Sudoku</a> puzzle. You should find that you never have to guess, but perhaps using a pencil and having a rubber to hand might be a good idea.</p>
<p>One more hint, colour in the blank squares, too – just use a different shade. This might sound obvious but leaving a square blank, once you have determined that it “must” be blank, might mean that it gets mistakenly shaded later. You can see in the figure above that I’ve have coloured the blank squares yellow, so we know that we should not make them black later.</p>
<h2>Getting mathematical</h2>
<p>If you don’t want to exercise the brain, you can get a computer to do it for you.</p>
<p>Jean Francois Puget presents one such – <strong>SPOILER ALERT: the next link reveals the solution</strong> – <a href="https://www.ibm.com/developerworks/community/blogs/jfp/entry/Solving_The_GCHQ_Christmas_Puzzle_As_A_MIP_With_Python?lang=en">methodology</a>, which is based on <a href="https://en.wikipedia.org/wiki/Integer_programming">mixed integer programming</a>.</p>
<p>You define the problem using variables, zeroes for white cells and ones for black cells. You next define constraints. For example, each row/column has to have the correct sequence of blacks cells in each row/column, separated by at least one white cell. You also need to define an objective function. This is usually something you are trying to minimise (for example, waste) or maximise (for example, profit). In the case of this puzzle, there is nothing to minimise or maximise, as once we have a valid solution we cannot improve on it.</p>
<p>Once we have defined the variables, constraints and objection function, we can hand it over to one of the many solvers that are available online and it will return the solution.</p>
<p>The downside of mathematical approaches to complex problems is that there may be a solution, but it could take <a href="https://theconversation.com/explainer-what-is-the-maths-behind-an-exam-timetable-22090">millions of years</a> to find it. Fortunately, this puzzle can be solved quickly.</p>
<h2>Other approaches</h2>
<p>If you don’t fancy either of the above two approaches, there are many other options. In a <a href="https://theconversation.com/how-to-get-ants-to-solve-a-chess-problem-22282">previous article</a> we discussed how ants could be used to solve chess puzzles. So if ants can play chess, they could certainly solve the GCHQ puzzle.</p>
<p>Whether it is worth the effort to develop the computer model required, however, is open to debate. The same could also be argued for the many other <a href="http://www.springer.com/gp/book/9781461469391">meta-heuristic approaches</a>. Almost any of them could solve this puzzle, but is it worth the development effort?</p>
<p>The puzzle has generated a lot of media interest and many people are trying to solve it. As we have shown above, there are already solutions on the internet and there is even more information about the subsequent puzzles on <a href="https://www.reddit.com/r/puzzles/comments/3w6ja9/gchqs_christmas_puzzler_thread_spoilers_tagged/">Reddit</a>. That does seem to go against the spirit of the puzzle, however, and the spirit of the season. Why not just print the grid, get out a pen and exercise the grey matter sometime over the festive period? You may even win the honest way.</p><img src="https://counter.theconversation.com/content/52401/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Graham Kendall does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Have you got the keen eye and quizzical mind of a professional spy? Let’s find out …Graham Kendall, Professor of Computer Science and Provost/CEO/PVC, University of NottinghamLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/513592015-11-27T13:42:24Z2015-11-27T13:42:24ZHow to solve a Rubik’s cube in five seconds<figure><img src="https://images.theconversation.com/files/103380/original/image-20151126-28303-18sdw23.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/theilr/345056969">theilr</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>This week, 14-year-old Lucas Etter set a new world record for solving the classic Rubik’s cube in Clarksville, Maryland, in the US, solving the scrambled cube in <a href="http://www.guinnessworldrecords.com/news/2015/11/confirmed-teenager-lucas-etter-sets-new-fastest-time-to-solve-a-rubiks-cube-wor">an astonishing 4.904 seconds</a>.</p>
<p>The maximum number of face turns needed to solve the classic Rubik’s cube, one that is segmented into squares laid out 3x3 on each face, is 20, and the maximum number of quarter turns is 26. It took 30 years to discover these numbers, which were <a href="http://cube20.org/">finally proved</a> by Tomas Rokicki and Morley Davidson using a mixture of mathematics and computer calculation. The puzzle does have 43,252,003,274,489,856,000 (43 times 10<sup>18,</sup> or 43 quintillion) possible configurations after all. </p>
<p>So how do the likes of Lucas Etter work out how to solve Rubik’s cube so quickly? They could read instructions, but that rather spoils the fun. If you want to work out how to do it yourself, you need to develop cube-solving tools. In this sense, a tool is a short sequence of turns which results in only a few of the individual squares on the cube’s faces changing position. When you have discovered and memorised enough tools, you can execute them one after the other in order as required to return the cube to its pristine, solved condition.</p>
<p>These tools require experimentation to discover. Here’s how I did it myself: go on holiday with a Rubik’s cube and a screwdriver. Do experiments to find tools. The trouble is that most experiments just scramble the cube horribly and you forget what you did so you cannot undo your moves. </p>
<p>Now you have a choice, either buy another Rubik’s cube, or take out your trusty screwdriver. Turn one face through 45 degrees, and place the screwdriver under a central piece of the rotated face. Using the screwdriver as a lever to gently prise it out, it’s then easy to take the cube apart completely and reassemble it in pristine form. </p>
<p>The final move of reassembly will be the reverse of the screwdriver trick: rotate one face 45 degrees and apply gentle pressure to put the final piece back in place.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">It’s a common problem.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/tangi_bertin/2445931396">tangi_bertin</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Sequences of moves of a cube form something that mathematicians call a
group. If <em>A</em> is a sequence of moves, then let <em>A<sup>-1</sup></em> (that’s “A inverse”) be the same sequence of moves performed in reverse. So if you perform <em>A</em> and then <em>A<sup>-1</sup></em>, the cube will be in the same state as was it when you began. The same is true if you first perform <em>A<sup>-1</sup></em> followed by <em>A</em>. </p>
<p>Now suppose that <em>B</em> is another sequence of moves. Many tools have the form of what mathematicians call a commutator: do <em>A</em>, then <em>B</em>, then <em>A<sup>-1</sup></em> and finally <em>B<sup>-1</sup></em>. If <em>A</em> and <em>B</em> commute, so that performing <em>A</em> then <em>B</em> is the same as doing <em>B</em> then <em>A</em>, then the commutator does nothing. From a mathematical point of view, a commutator measures failure to commute, and is a key notion in group theory. When I had a Rubik’s cube in one hand, and a screwdriver in the other, it was natural to look at how commutators behave.</p>
<p>Think of the overall structure of the different configurations of a Rubik’s cube as a labyrinth, which has that many chambers, each of which contains a Rubik’s cube in the state which corresponds to that chamber. From each chamber there are 12 doors leading to other chambers, each door corresponding to a quarter turn of one of the six faces of a cube. The type of turn needed to pass through each door is written above it, so you know which door is which. Your job is to navigate your way from a particular chamber to the one where the cube on the table is in perfect condition.</p>
<p>The tools that you have discovered are ways of getting nearer to the goal. So you don’t need to plan your route in advance, you just execute the rotations of each tool so that you get steadily closer to and finally reach the winning chamber. The mathematical result in Rokicki and Davidson’s paper shows that, no matter where you are in the labyrinth, it’s possible to reach the winning chamber by passing through at most 26 doors – although the route you find using your tools is not likely to be that efficient.</p>
<p>How to put this to use to solve the cube in five seconds? Someone like young Lucas Etta who is interested in speed solutions will not only have memorised a large number of tools, they’ll also have practised them until they can perform it very quickly. This is mostly a matter of dexterity and practice, but it’s also important to have a high-quality cube that can be manipulated smoothly and with great precision.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/0RfJbcydNJ0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Others, rather than going for speed, develop the skill of solving Rubik’s cube while blindfolded or with the cube held behind their back. In the competitive version of this variation, the solver is given a limited amount of time to study the scrambled cube and plan their solution, before they have to carry out their solution from memory without looking at the cube again. </p>
<p>In terms of our metaphor of a labyrinth, this corresponds to all the Rubik’s cubes in all the chambers being removed, except for the one on which you start. You can’t take that cube with you, but you can study it carefully and plan your whole route to the winning chamber in advance. Quite a feat of memory, and not for those with just a passing interest in the cube.</p><img src="https://counter.theconversation.com/content/51359/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Geoff Smith does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Don’t worry – there are only 43,252,003,274,489,856,000 configurations.Geoff Smith, Senior Lecturer in Mathematics, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/430532015-06-25T10:36:38Z2015-06-25T10:36:38ZDon’t freak if you can’t solve a math problem that’s gone viral<figure><img src="https://images.theconversation.com/files/86322/original/image-20150624-31526-1jbqvaz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Are you smarter than a third grader in Vietnam?</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=252676549&src=lb-29877982">Woman image via www.shutterstock.com</a></span></figcaption></figure><p>It’s been quite a year for mathematics problems on the internet. In the last few months, three questions have been online everywhere, causing consternation and head-scratching and blowing the minds of adults worldwide as examples of what kids are expected to know these days.</p>
<p>As a mathematician, I suppose I should subscribe to the “no such thing as bad publicity” theory, except that problems of this ilk a) usually aren’t that difficult once you get the trick, b) sometimes aren’t even math problems and c) fuel the defeatist “I’m not good at math” fire that pervades American culture. The inability to solve such a problem quickly is certainly not indicative of a person’s overall math skill, nor should it prompt a crisis of confidence about the state of American math aptitude.</p>
<h2>When is Cheryl’s birthday?</h2>
<p>In April, the internet erupted with shock that 10-year-olds in Singapore were asked to answer the following question on an exam.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=436&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=436&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=436&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=548&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=548&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=548&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The logic puzzle from the Singapore and Asian Math Olympiads.</span>
</figcaption>
</figure>
<p>Except that it wasn’t for elementary school students at all; rather it appeared on an Asian Olympiad exam designed for mathematically talented high school students. What’s more, this isn’t even a math problem, but a logic problem. It’s true that students tend to learn formal logic via mathematics (plane geometry in particular), so it is common to see problems of this type in mathematics competitions. When I was in junior high, we spent a good deal of time on these puzzles in my language arts class, and I met them again when taking the GRE prior to entering graduate school (the test contains a whole section of them). </p>
<p>If you’re stumped, check out a <a href="http://www.independent.co.uk/news/world/asia/singapore-maths-problem-can-you-solve-the-viral-maths-question-that-was-set-to-children-in-singapore-10173090.html">solution to the problem</a>.</p>
<h2>Vietnamese eight-year-olds do arithmetic</h2>
<p>A month later, we heard about a third grade teacher in Vietnam who set the following puzzle for his students. Place the digits from 1 to 9 in this grid, using each only once (the : represents division).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=433&fit=crop&dpr=1 600w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=433&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=433&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=545&fit=crop&dpr=1 754w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=545&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=545&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A puzzle for Vietnamese children.</span>
<span class="attribution"><span class="source">VN Express</span></span>
</figcaption>
</figure>
<p>This reminds me of the (probably apocraphyl) story of one of the greatest mathematicians in history, <a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss">Carl Friedrich Gauss</a>. Legend has it that when Gauss was seven or eight, his teacher, wanting to occupy his students for a while, told the class to add up the numbers from 1 to 100. Gauss thought about it for 30 seconds or so and wrote the correct answer, 5,050, on his slate and turned it in.</p>
<p>The puzzle above has a similar feel. It’s really a question about knowing the order of arithmetic operations (multiplication/division, addition/subtraction, in that order). Beyond that, it just takes trial and error; that is, it’s kind of just busy work. Someone who knows some algebra might be able to generate some equations to gain insight into how you might find a <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/may/20/can-you-do-the-maths-puzzle-for-vietnamese-eight-year-olds-that-has-stumped-parents-and-teachers">solution</a>.</p>
<p>Another approach would be to open up a spreadsheet program and just try all the possibilities. Since there are nine choices for the first box, then eight choices for the second, and so on, there are only (9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880 possible configurations, of which only a few will give a valid equation. This can be programmed with very little effort.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=603&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Yellow or orange, students didn’t find the problem sweet.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-275334341/stock-photo-orange-and-yellow-jelly-candies-closeup-sweet-background.html">Candy image via www.shutterstock.com</a></span>
</figcaption>
</figure>
<h2>Hannah’s sweets</h2>
<p>Just a couple of weeks ago, students in the UK vented their frustration via social media about a problem on the Edexcel GCSE (General Certificates of Secondary Education) mathematics exam. It is a probability question: Hannah has a bag containing <em>n</em> candies, six of which are orange and the rest of which are yellow. She takes two candies out of the bag and eats them. The probability that she ate two orange candies is 1/3. Given this, show that <em>n² - n - 90 = 0</em>. The students’ complaint? It’s too difficult.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"606837485239463937"}"></div></p>
<p>I’ve taught math long enough to recognize the pitfalls of setting this problem. The students actually have the knowledge to do it, if they know basic probability, but it is unlike problems they would have practiced. A typical question would indicate the total number of candies in the bag and ask students to compute the probability of a certain outcome. This question gives the probability and asks for a condition on the number of candies. It’s just algebra. You may read the solution (and some humorous memes about the question) <a href="http://www.telegraph.co.uk/education/11652918/Students-vent-their-frustration-at-Edexcel-GCSE-maths-exam.html">here</a>.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">What does his lifelong future with math look like?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/grahams__flickr/360774920">Prisoner 5413</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span>
</figcaption>
</figure>
<h2>A nation at risk?</h2>
<p>Mathematicians dread cocktail parties because we inevitably have to endure the response we receive when asked what we do: “Oh, I hated (or am terrible at) math.” No other subject in school receives such scorn, nor would we find it acceptable for an adult to admit they are terrible at reading or writing. So when these “unsolvable” problems pop up, they simply reinforce our culture’s math anxiety. </p>
<p>And that’s a real shame, because everyone likes math when they’re young. We all like to count. We like playing with blocks and shapes. We all use math daily whether we realize it or not – reading maps, planning routes, calculating tips. I once had a flooring installer tell me he was bad at math <em>while I watched him lay tile</em>. <a href="http://www.theatlantic.com/education/archive/2013/10/the-myth-of-im-bad-at-math/280914/">It’s a myth</a> that all these people can’t do math. When people say they are “bad at math,” they usually mean that they had trouble with algebra, although if you corner them and ask the right questions you can usually make them realize that they use algebra all the time without noticing it. This leads to <a href="https://grantwiggins.wordpress.com/2013/04/10/my-100th-post-so-why-not-bash-algebra/">valid criticisms</a> of how we teach math, but it doesn’t mean we’re a nation of math idiots.</p>
<p>So, the next time one of these outrageous problems comes along, instead of giving in to anxiety, why not think about it for a few minutes and try to find a solution? You might be surprised how satisfying it can be.</p><img src="https://counter.theconversation.com/content/43053/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kevin Knudson does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>People shouldn’t let these tricky puzzlers reinforce their misguided notion that they stink at math.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/402162015-04-16T01:58:18Z2015-04-16T01:58:18ZIt’s often the puzzles that baffle that go viral<figure><img src="https://images.theconversation.com/files/78033/original/image-20150415-24615-x1edmo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A love of puzzles.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/jypsygen/3732589905">Flickr/jypsygen</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Many of us are devoted to our morning crossword, acrostic, anagram or Sudoku puzzle. Quite a few religiously listen to the Sunday <a href="http://www.npr.org/people/2101852/will-shortz">Puzzlemaster</a> <a href="http://willshortz.com/">Will Shortz</a> (who also sets puzzles for the New York Times) on National Public Radio.</p>
<p>So perhaps it is not surprising – even though many of us did not like school maths – that every so often a logical puzzle or maths problem goes viral. The most recent example is “<a href="http://www.9news.com.au/world/2015/04/14/11/09/can-you-solve-this-singaporean-maths-problem">Cheryl’s birthday</a>”.</p>
<p>The puzzle was originally posted on the <a href="https://www.facebook.com/kennethjianwenz/photos/a.173663129479243.1073741827.167504136761809/385751114937109/">Facebook page</a> of Singapore media personality Kenneth Hong, who said it was causing some debate with his wife.</p>
<blockquote>
<p>Albert and Bernard just became friends with Cheryl. and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.</p>
<pre class="highlight plaintext"><code> May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17
</code></pre>
<p>Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.</p>
<p>Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.</p>
<p>Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.</p>
<p>Albert: Then I also know when Cheryl’s birthday is.</p>
<p>So when is Cheryl’s birthday?</p>
</blockquote>
<p>Many people have tried their hand at solving the puzzle, including mathematician and writer <a href="http://www.alexbellos.com/">Alex Bellos</a>. Alex runs through it <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/13/how-to-solve-albert-bernard-and-cheryls-birthday-maths-problem">line by line</a>, showing how he gets to the solution: the key is to ask what each of Bernard and Albert learn from the other’s statements.</p>
<p>Knowing what information is superfluous is often helpful, explains Alex:</p>
<blockquote>
<p>The only way that Bernard could know the date with a single number, however, would be if Cheryl had told him 18 or 19, since of the ten date options only these numbers appear once, as May 19 and June 18.</p>
</blockquote>
<p>Proceeding in like fashion (read the rest of Alex’s explanation <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/13/how-to-solve-albert-bernard-and-cheryls-birthday-maths-problem">here</a>) we are led to:</p>
<blockquote>
<p>The answer, therefore is July 16.</p>
</blockquote>
<p>It has been suggested that it would have been easier to consult Cheryl’s Facebook page!</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"587846560685297665"}"></div></p>
<h2>Where did Cheryl’s birthday problem arise?</h2>
<p>The problem was set as a hard question on a regional competition for bright high school students: the Singapore and Asian Schools Math Olympiad (<a href="http://mathsolympiads.org/">SASMO</a>).</p>
<p>There is another in <a href="APSMO%20Maths%20Olympiad">Australasia</a>, and they culminate in the annual <a href="https://www.imo-official.org/">International Mathematical Olympiad</a>, with the questions getting progressively harder as the regions grow. The world champion Maths Olympians (or “mathletes”) are incredibly gifted young problem solvers.</p>
<p>Cheryl’s birthday problem was inadvertently originally described as a grade 5 question for ordinary school kids in Singapore. This probably helped it go viral.</p>
<p>But why do certain things go viral in the first place? By convenient coincidence a new scientific study on <a href="http://www.scientificamerican.com/article/the-secret-to-online-success-what-makes-content-go-viral/">The Secret to Online Success: What Makes Content Go Viral</a> has just appeared – as described this week in Scientific American.</p>
<p>The researchers looked at what it was that made certain things “spread like wildfire” and whether it was possible to deliberately make content that would make something achieve viral status.</p>
<p>They found there were a number of things that could increase the chances of any content being widely shared.</p>
<blockquote>
<p>Make it emotional – ideally triggering emotions like anger, anxiety or awe that tend to make our hearts race; and if you can, make it positive. This may be more even effective than other methods that are currently in wide use like targeting “influentials,” or opinion leaders. Crafting contagious content, as this research suggests, may provide more bang for your buck and create more reliably viral content.</p>
</blockquote>
<p>You can tick off the emotional positives that helped Cheryl’s birthday problem go viral. It was unlikely and curious, but reassuring and nonthreatening. It did not tell you you were a dummy if you could not figure it out, far from it! A media personality helped it get going, and so on.</p>
<p>You can also inoculate against things going viral. My own worst-read blog in the Conversation had the unappetising title <a href="https://theconversation.com/danger-of-death-are-we-programmed-to-miscalculate-risk-4598">Danger of death: are we programmed to miscalculate risk?</a>. This seemed to be bringing only bad news when in fact we discussed how to calculate relative risk better and be less alarmed.</p>
<p>A <a href="http://www.mathresources.com/">software company</a> I helped set up 20 years ago saw the sales of one product sky-rocket when the name was changed from MathProbe (too medical?) to the more inviting Let’s do Math.</p>
<h2>What makes a puzzle easy or hard</h2>
<p>Cultural differences matter. For many years students in the French South Pacific were given exactly the same mathematics exams as those in Paris or Bordeaux.</p>
<p>A famous question asked students to “consider a dairy cow looking at the pole star on a snowy night”. This was served up for kids in the French South Pacific who in a far-ago pre-internet world had never seen snow, the northern sky, or even a cow.</p>
<p>Likewise numeric puzzles, such as Sudoku or nonograms, often arise in Japan or Korea whose ideogram based scripts make crosswords and anagrams a non-starter.</p>
<p>Keith Devlin, a mathematician who is a very gifted expositor, has a book <a href="http://www.amazon.com/Math-Instinct-Mathematical-Genius-Lobsters/dp/1560256729/ref=la_B000APRPC6_1_12?s=books&ie=UTF8&qid=1429061413&sr=1-12">The Math Instinct</a> in which, among other things he describes how a change in language can make a seeming impervious problem easy.</p>
<p>For example, doing arithmetic in bases other than base ten sounds formidable. But whenever you watch a 50-50 cricket match and read on the screen that it is over 42.3, the ‘42’ are in base ten and the ‘3’ is in base six. Easy-peasy?</p>
<p>Similarly, you might discover that an abstract problem about probability can be expressed in terms of <a href="https://theconversation.com/how-betting-works-and-why-the-melbourne-cup-skews-the-odds-33357">horse races</a> or lotteries, or something else you know lots about.</p>
<h2>Some other problems that went viral (or should have)</h2>
<p>Conditional, sometimes counter-factual, thinking of the kind needed to determine Cheryl’s birthday is not something most humans find easy. Though there is a large subset of humanity who find some or all puzzles both enticing and accessible.</p>
<p>I had a friend who could usually do the notoriously hard cryptic crossword in the London Times in about ten minutes. He could not really explain how, he just saw the answers. I have another friend who is an expert crossword puzzle setter, as of course is Will Shortz. </p>
<p>Setting good puzzles or just good maths exam questions is an art in itself.</p>
<p>One of the most popular viral puzzles is known as the Monty Hall or “three door problem”. It has already been explained in <a href="https://theconversation.com/the-monty-hall-problem-going-with-your-gut-will-get-your-goat-14195">The Conversation</a> and makes a surprising story. It first went viral in large part because the correct answer seemed so unintuitive even to professional logicians.</p>
<p>To those who want to follow up on some other mind tickling examples I conclude with:</p>
<ol>
<li><p>The paradox of the <a href="http://docserver.carma.newcastle.edu.au/209/">unexpected hanging</a> and its gentler version the <a href="http://www-math.mit.edu/%7Etchow/unexpected.pdf">surprise examination</a> paradox.</p></li>
<li><p>The paradox of the <a href="https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/">blue eyed islanders</a>.</p></li>
</ol><img src="https://counter.theconversation.com/content/40216/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the Australian Research Council.</span></em></p>Have you heard the one about Cheryl’s birthday? It’s the latest puzzle that’s baffling people across the world.Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/52342012-02-12T11:29:12Z2012-02-12T11:29:12ZGood at Sudoku? Here’s some you’ll never complete<figure><img src="https://images.theconversation.com/files/7536/original/c4p43n9v-1328839438.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">There's far more to the popular maths puzzle than putting numbers in a box.</span> <span class="attribution"><span class="source">zlovall</span></span></figcaption></figure><p>Last month, a team led by Gary McGuire from University College Dublin in Ireland made <a href="http://www.ucd.ie/news/2012/01JAN12/100111-There-is-no-16-clue-or-less-Sudoku-mathematician-proves.html">an announcement</a>: they had proven you can’t have a solvable Sudoku puzzle with less than 17 numbers already filled in.</p>
<p>Unlike most mathematical announcements, this was quickly picked up by the popular scientific media. Within a few days, the new finding had been <a href="http://www.nature.com/news/mathematician-claims-breakthrough-in-sudoku-puzzle-1.9751">announced in Nature</a> and other outlets.</p>
<p>So where did this problem come from and why is its resolution interesting?</p>
<p>As you probably know, the aim of a Sudoku puzzle is to complete a partially-filled nine-by-nine grid of numbers. There are some guidelines: the numbers one to nine must appear exactly once each in every row, column and three-by-three sub-grid. </p>
<p>As with a crossword, a valid Sudoku puzzle must have a unique solution. There’s only one way to go from the initial configuration (with some numbers already filled in) to a completed grid.</p>
<p>Newspapers often grade their puzzles as easy, medium or hard, which will depend on how easy it is at every stage of solving the puzzle to fill in the “next” number. While a puzzle with a huge number of initial clues will usually be easy, it is not necessarily the case that a puzzle with few initial clues is difficult. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/7533/original/3g6pkfzg-1328838628.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Reckon you can complete a 17-clue Sudoku puzzle? (answer below)</span>
<span class="attribution"><span class="source">Gordon Royle</span></span>
</figcaption>
</figure>
<p>When Sudoku-mania swept the globe in the mid-2000s, many mathematicians, programmers and computer scientists – amateur and professional – started to investigate Sudoku itself. They were less interested in solving individual puzzles, and more focused on asking and answering mathematical and/or computational questions about the entire universe of Sudoku puzzles and solutions.</p>
<p>As a mathematician specialising in the area of <a href="http://en.wikipedia.org/wiki/Combinatorics">combinatorics</a> (which can very loosely be defined as the mathematics of counting configurations and patterns), I was drawn to combinatorial questions about Sudoku. </p>
<p>I was particularly interested in the question of the smallest number of clues possible in a valid puzzle (that is, a puzzle with a unique solution). </p>
<p>In early 2005, I found a handful of 17-clue puzzles on a long-since forgotten Japanese-language website. By slightly altering these initial puzzles, I found a few more, then more, and gradually built up a “library” of 17-clue Sudoku puzzles which I made available online at the time. </p>
<p>Other people started to send me their 17-clue puzzles and I added any new ones to the list until, after a few years, I had collected more than 49,000 different 17-clue Sudoku puzzles.</p>
<p>By this time, new ones were few and far between, and I was convinced we had found almost all of the 17-clue puzzles. I was also convinced there was no 16-clue puzzle. I thought that demonstrating this would either require some new theoretical insight or clever programming combined with massive computational power, or both. </p>
<p>Either way, I thought proving the non-existence of a 16-clue puzzle was likely to be too difficult a challenge.</p>
<p>They key to McGuire’s approach was to tackle the problem indirectly. The total number of completed puzzles (that is, completely filled-in grids) is astronomical – 5,472,730,538 – and trying to test each of these to see if any choice of 16 cells from the completed grid forms a valid puzzle is far too time-consuming.</p>
<p>Instead, McGuire and colleagues used a different, indirect approach. </p>
<p>An “unavoidable set” in a completed Sudoku grid is a subset of the clues whose entries can be rearranged to leave another valid completed Sudoku grid. For a puzzle to be uniquely completable, it must contain at least one entry from every unavoidable set.</p>
<p>See the picture below to see what I mean.</p>
<p>If a completed grid contains the ten-clue configuration in the left picture, then any valid Sudoku puzzle must contain at least one of those ten clues. If it did not, then in any completed puzzle, those ten positions could either contain the left-hand configuration or the right-hand configuration and so the solution would not be unique.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=272&fit=crop&dpr=1 600w, https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=272&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=272&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=342&fit=crop&dpr=1 754w, https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=342&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/7535/original/2hyyyh3c-1328839215.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=342&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">Gordon Royle</span></span>
</figcaption>
</figure>
<p>While finding all the unavoidable sets in a given grid is difficult, it’s only necessary to find enough unavoidable sets to show that no 16 clues can “hit” them all. In the process of resolving this question, McGuire’s team developed new techniques for solving the <a href="http://www.shannarasite.org/kb/kbse42.html">“hitting set”</a> problem. </p>
<p>It’s a problem that has many other applications – any situation in which a small set of resources must be allocated while still ensuring that all needs are met by at least one of the selected resources (i.e. “hit”) can be modelled as a hitting set problem.</p>
<p>Once the theory and software was in place, it was then a matter of running the programs for each of the 5.5 billion completed grids. As you can imagine, this required substantial computing power. </p>
<p>After 7 million core-CPU hours on a supercomputer (the equivalent of a single computer running for 7 million hours) and a year of actual elapsed time, the result was announced a few weeks ago, on New Year’s Day. </p>
<p>So is it correct?</p>
<p>The results of any huge computation should be evaluated with some caution, if not outright suspicion, especially when the answer is simply “no, doesn’t exist”, because there are many possible sources of error.</p>
<p>But in this case, I feel the result is far more likely to be correct than otherwise, and I expect it to be independently-verified before too long. In addition, McGuire’s team built on many different ideas, discussions and computer programs that were thrashed out between interested contributors to various online forums devoted to the mathematics of Sudoku. In this respect, many of the basic components of their work have already been thoroughly tested.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=599&fit=crop&dpr=1 600w, https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=599&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=599&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=753&fit=crop&dpr=1 754w, https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=753&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/7534/original/zy7sj894-1328838628.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=753&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Solution to the 17-clue Sudoku puzzle, above.</span>
<span class="attribution"><span class="source">Gordon Royle</span></span>
</figcaption>
</figure>
<p>And so back to the question: why is the resolution of this problem interesting? And is it important?</p>
<p>Certainly, knowing that the smallest Sudoku puzzles have 17 clues is not in itself important. But the immense popularity of Sudoku meant that this question was popularised in a way that many similar questions have never been, and so it took on a special role as a “challenge question” testing the limits of human knowledge.</p>
<p>The school students to whom I often give outreach talks have no real concept of the limitations of computers and mathematics. In my past talks, these students were almost always astonished to know that the answer to such a simple question was just not known.</p>
<p>And now, in my future outreach talks, I will describe how online collaboration, theoretical development and significant computational power were combined to solve this problem, and how this process promises to play an increasing role in the future development of mathematics.</p><img src="https://counter.theconversation.com/content/5234/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Gordon Royle does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Last month, a team led by Gary McGuire from University College Dublin in Ireland made an announcement: they had proven you can’t have a solvable Sudoku puzzle with less than 17 numbers already filled in…Gordon Royle, Professor of Mathematics, The University of Western AustraliaLicensed as Creative Commons – attribution, no derivatives.