tag:theconversation.com,2011:/institutions/harvey-mudd-college-2034/articlesHarvey Mudd College2017-08-02T01:20:25Ztag: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/414222015-06-08T10:16:56Z2015-06-08T10:16:56ZClosing the computer science gender gap: how one woman is making a difference in many lives<figure><img src="https://images.theconversation.com/files/83895/original/image-20150604-2959-ek9mgz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Maria Klawe: Changing the dynamics of a male-dominated field.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/shuwu/521134573/in/photolist-N3X9R-dgsxPC-gmrD9f-tsEv2S-sw7BHr-sBuBBr-tt2aAV-dgsxQT-dgsyzr-dgsAbA-dgsBvi-dgsy4U-dgsBDo-N3PY1-N3WKt-N3P6C-N3Xj6-N3Muu-N3Qhy-N3XvP-N3Wgi-N3Y68-N3N2j-guN32v-guMskb-dgsyLo-dgszXC-dgszhk-dgszAT-dgsA3R-dgsD4N-dgsB9i-dgszqd-dgsATy-dgsAxU-dgsC5A-dgsAXg-dgswNK-dgsCqL-dgszby-dgsy1D-dgswyZ-dgsxdZ-o4ecGV-onsEAB-o4dacZ-o4dmVD-oiF2vL-okv7Lw-okF2tA">Shu Wu</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>I’ve been passionate about increasing women’s participation in computer science for more than 25 years. While the number of undergraduate women pursuing some STEM (science, technology, engineering, math) fields like biology and chemistry has steadily <a href="http://nsf.gov/statistics/2015/nsf15311/start.cfm">increased</a> over the past couple of decades, women’s participation in computer science (CS) has actually been <a href="http://nsf.gov/statistics/2015/nsf15311/digest/theme2.cfm#compsci">declining</a>.</p>
<p>Indeed, within the last 20 years the percentage of undergraduate women who received CS degrees plummeted by almost 40%. According to the <a href="http://www.nsf.gov/">National Science Foundation</a>, in 1995, 29% of bachelor’s degrees in CS were awarded to women; by 2012, the most recent year for which NSF data is available, only <a href="http://www.nsf.gov/statistics/2015/nsf15311/tables/pdf/tab5-1.pdf">18%</a> of CS undergraduates were women. </p>
<p>The <a href="http://cra.org/">Computing Research Association (CRA)</a> reports even lower numbers. CRA’s annual <a href="http://cra.org/resources/taulbee/">Taulbee Survey</a> of over 100 major computer science departments in North America found that by the mid-2000s, the percentage of women graduating with CS bachelor’s degrees averaged <a href="http://archive.cra.org/statistics/survey/0506.pdf">14%.</a> </p>
<p>For me this issue is both personal and global.</p>
<h2>Working to remove barriers</h2>
<p>I’m a mathematician and a computer scientist. Back when I was getting my PhD in math, there were very few women in my field. Despite receiving discouragement because of my gender, I worked hard. </p>
<p>And I excelled. I went on to take up leadership roles in many places – at IBM, University of British Columbia, Princeton and now at Harvey Mudd. I’ve been the first woman in nearly all these positions.</p>
<p>I know the struggles that can hinder women when they are working in a <a href="http://nsf.gov/statistics/2015/nsf15311/digest/theme5.cfm#women">predominantly male field</a>. I also know firsthand how computer science and technology make for a great career, offering a <a href="http://www.payscale.com/research/US/Degree=Bachelor_of_Science_%28BS_%2F_BSc%29,_Computer_Science_%28CS%29/Salary">good income</a>, work-life balance and opportunities to travel. They also offer a chance to make significant contributions to the world, by working on important societal problems. </p>
<p>I want young women to have these opportunities. </p>
<p>I’ve been working on this issue for decades. When I came to Harvey Mudd College in 2006, the CS department was averaging only about 10% women majors. The faculty had decided to make significant changes to attract more women. </p>
<p>They redesigned their introductory computer science courses to focus less on straight programming and more on creative problem-solving. They included topics to show the breadth of the field and the ways in which it could benefit society.</p>
<p>In order to reduce the intimidation factor for women and other students with no prior coding experience, they split the course into two sections, black and gold (Harvey Mudd’s colors), with black for those who had prior programming experience and gold for those with no prior experience. </p>
<p>This worked wonders to create a supportive atmosphere. </p>
<h2>Making the field exciting for women</h2>
<p>Instead of traditional homework, which can be isolating, the faculty assigned team-based projects so that students coded together. And most importantly, they made the courses fun. The intro CS courses went from being the least-liked course in our core curriculum to being the most popular.</p>
<p>After the courses were introduced in 2007, we saw an immediate and steady increase in the percentage of female students majoring in CS. Within four years, we went from averaging around 10% women majors to averaging 40%. We have continued to average 40% since 2011. </p>
<p>In addition, faculty created early summer research opportunities designed for students who had completed only one or two CS courses, and encouraged their first-year female students to participate. A <a href="http://www.bhef.com/sites/g/files/g829556/f/201305/report_2013_FinalInterventionsResearch.pdf">number of studies</a> have shown that research experiences for undergraduate students increase retention and confidence in STEM fields, factors that are <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2879378/">particularly important for women and minorities</a>. </p>
<p>Harvey Mudd’s female students who participated in early CS research projects indeed reported greatly increased interest in the discipline and a boost in confidence. They realized they could do the work of a computer scientist and that they enjoyed it as well. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/83891/original/image-20150604-2927-1shj1ak.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">Even today, there aren’t enough women entering the field of computer science.</span>
<span class="attribution"><span class="source">Harvey Mudd College</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>We also send large contingents of women students each year to the <a href="http://gracehopper.org/">Grace Hopper Celebration of Women in Computing,</a> the largest conference for women working in technology fields. At this event, students get to see role models and are excited about the many amazing technology career paths they can pursue. </p>
<p>Other institutions are starting to take up our approaches.</p>
<p>For instance, the <a href="https://www.hmc.edu/about-hmc/2014/09/24/harvey-mudd-launches-initiative-increase-diversity-computer-science/">Building Recruiting And Inclusion for Diversity (BRAID) initiative</a> that we launched with the <a href="http://anitaborg.org/">Anita Borg Institute</a> is working to build computer science diversity at 15 academic institutions. We are about to offer our intro computer science course as a <a href="https://www.edx.org/school/harveymuddx">free MOOC on edX</a>, so that professors and students can access the course materials.</p>
<h2>There’s still work to be done</h2>
<p>However, many barriers and challenges remain. </p>
<p><a href="http://nsf.gov/statistics/2015/nsf15311/start.cfm">The National Science Foundation’s most recent (2012) report</a> shows that computer science has the lowest proportion of women receiving bachelor degrees of all the STEM fields. The percentage dipped to a mere <a href="http://cra.org/uploads/documents/resources/taulbee/0809.pdf">11%</a> from 2007-2009 and gradually returned to <a href="http://cra.org//uploads/documents/resources/crndocs/2014-Taulbee-Survey.pdf">14%</a> by 2013-2014. </p>
<p>Computer science should be a required part of secondary education, but a lot of schools don’t have computer science teachers. Most young people who go to college today have <a href="http://www.exploringcs.org/resources/cs-statistics">not had much exposure to computer science</a>. </p>
<p>We also have to combat the cultural belief that some people are simply born with math, science or computer talent and others are simply “not good at it.” There’s <a href="http://www.ibparticipation.org/pdf/Designing_for_Success.pdf">lots of research</a> that shows that persistence and hard work play a much larger role in success in any area of science and engineering than “native ability.” Another serious challenge is posed by the media portrayal of careers in technology, which builds certain stereotypes.</p>
<p>I want people to think about how we can change our images of who we consider to be competent in technology. At the moment, the image of the computer scientist is limited to a white or Asian male. </p>
<p>One thing we know for sure is that you get better solutions if you have more diverse teams working on them. We need the female perspective to get the best solutions to very pressing problems. </p>
<h2>Need for diversity</h2>
<p>We also need more African Americans, Latinos/Latinas, poets, football players and artists involved in creating technology. Right now there is unfilled demand for computer science grads and not just in the tech industry.</p>
<p>I want computer science and technology to be a world that embraces everyone who has passion, ability and interest, whether they look like the dominant group or not. </p>
<p>Today, computer science touches all industries. Its products are embedded in our daily lives. Addressing the significant problems of the world – from climate change to health care to poverty – will involve technology. </p>
<p>I think the world will be an incredibly exciting place and we will see amazing technological developments when we create a much more diverse tech community.</p><img src="https://counter.theconversation.com/content/41422/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Maria Klawe serves on the boards of Microsoft Corporation and Broadcom Corporation, as well as the nonprofits Math for America and EdReports. Klawe is a past president of the Association of Computing Machinery in New York, past chair of the board of trustees of the Anita Borg Institute for Women and Technology in Palo Alto, and a past trustee of the Institute for Pure and Applied Mathematics in Los Angeles. She has held leadership positions with the American Mathematical Society, the Computing Research Association, the Society for Industrial and Applied Mathematics and the Canadian Mathematical Society.She is a trustee for the Mathematical Sciences Research Institute in Berkeley. Until recently, she was a member of the Stanford Engineering Advisory Council and the Advisory Council for the Computer Science Teachers Association. She was elected as a fellow of the Association of Computing Machinery in 1996 and as a founding fellow of the Canadian Information Processing Society in 2006.
Harvey Mudd College receives funding from many organizations, including The Ahmanson Foundation, The Annenberg Foundation, The Arnold and Mabel Beckman Foundation, The Arthur Vining Davis Foundations, Camille & Henry Dreyfus Foundation, Edison International, The Hearst Foundations, The Howard Hughes Medical Institute, Honeywell, Intel Corporation, The Fletcher Jones Foundation, The Andrew W. Mellon Foundation, The National Institute of Standards & Technology, The National Institutes of Health, The National Science Foundation, The Ralph M. Parsons Foundation, Research Corporation for Science Advancement, and The John Stauffer Charitable Trust.
</span></em></p>The number of women receiving undergraduate degrees in computer science has plummeted by 40% in the last 20 years. How can this be changed? Ask Maria Klawe.Maria Klawe, President, Harvey Mudd CollegeLicensed as Creative Commons – attribution, no derivatives.