{"id":756,"date":"2015-05-01T00:01:40","date_gmt":"2015-05-01T04:01:40","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=756"},"modified":"2015-05-01T14:22:12","modified_gmt":"2015-05-01T18:22:12","slug":"famous-unsolved-math-problems-as-homework","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2015\/05\/01\/famous-unsolved-math-problems-as-homework\/","title":{"rendered":"Famous Unsolved Math Problems as Homework"},"content":{"rendered":"

By Benjamin Braun, <\/i>Editor-in-Chief<\/i><\/a>, University of Kentucky<\/i><\/p>\n

One of my favorite assignments for students in undergraduate mathematics courses is to have them work on unsolved math problems. \u00a0An unsolved math problem, also known to mathematicians as an \u201copen\u201d problem, is a problem that no one on earth knows how to solve. \u00a0My favorite unsolved problems for students are simply stated ones that can be easily understood. \u00a0In this post, I\u2019ll share three such problems that I have used in my classes and discuss their impact on my students.<\/p>\n

Unsolved Problems<\/b><\/p>\n

The Collatz Conjecture<\/i><\/a>. Given a positive integer \\(n\\), if it is odd then calculate \\(3n+1\\). \u00a0If it is even, calculate \\(n\/2\\). \u00a0Repeat this process with the resulting value. For example, if you begin with \\(1\\), then you obtain the sequence \\[ 1,4,2,1,4,2,1,4,2,1,\\ldots \\] which will repeat forever in this way. \u00a0If you start with a \\(5\\), then you obtain the sequence \\(5,16,8,4,2,1,\\ldots\\), and now find yourself in the previous case. \u00a0The unsolved question about this process is: If you start from any positive integer, does this process always end by cycling through \\(1,4,2,1,4,2,1,\\ldots\\)? \u00a0Mathematicians believe that the answer is yes, though no one knows how to prove it. This conjecture is known as the Collatz Conjecture (among many other names), since it was first asked in 1937 by Lothar Collatz.<\/p>\n

The Erd<\/i>\u0151s-Strauss Conjecture<\/i><\/a>. A fascinating question about unit fractions is the following: For every positive integer \\(n\\) greater than or equal to \\(2\\), can you write \\(\\frac{4}{n}\\) as a sum of three positive unit fractions? \u00a0For example, for \\(n=3\\), we can write \\[\\frac{4}{3}=\\frac{1}{1}+\\frac{1}{6}+\\frac{1}{6} \\, . \\] \u00a0For \\(n=5\\), we can write \\[ \\frac{4}{5}=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{20} \\] or \\[\\frac{4}{5}=\\frac{1}{2}+\\frac{1}{5}+\\frac{1}{10} \\, . \\] \u00a0In other words, if \\(n\\geq 2\\) can you always solve the equation \\[ \\frac{4}{n}=\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\\] using positive integers \\(a\\), \\(b\\), and \\(c\\)? \u00a0Again, most mathematicians believe that the answer to this question is yes, but a proof remains elusive. \u00a0This question was first asked by Paul Erd\u0151s and Ernst Strauss in 1948, hence its name, and mathematicians have been working hard on it ever since.<\/p>\n

Lagarias\u2019s Elementary Version\u00a0of the Riemann Hypothesis<\/i><\/a>. \u00a0For a positive integer \\(n\\), let \\(\\sigma(n)\\) denote the sum of the positive integers that divide \\(n\\). \u00a0For example, \\(\\sigma(4)=1+2+4=7\\), and \\(\\sigma(6)=1+2+3+6=12\\). \u00a0Let \\(H_n\\) denote the \\(n\\)-th harmonic number, i.e. \\[ H_n=1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\cdots+\\frac{1}{n} \\, .\\] Our third unsolved problem is: Does the following inequality hold for all \\(n\\geq 1\\)? \\[ \\sigma(n)\\leq H_n+\\ln(H_n)e^{H_n} \\] In 2002, Jeffrey Lagarias proved that this\u00a0problem is equivalent to the Riemann Hypothesis<\/a>, a famous question about the complex roots of the Riemann zeta function. \u00a0Because it is equivalent to the Riemann Hypothesis, if you successfully answer it, then the Clay Mathematics Foundation will reward you with $1,000,000<\/a>. \u00a0While the statement of this problem is more complicated than the previous two, it doesn\u2019t involve anything beyond natural logs and exponentials at a precalculus level.<\/p>\n

Impact on Students<\/b><\/p>\n

I\u2019ve used all three of these problems, along with various others<\/a>, as the focus of in-class group work and as homework problems in undergraduate mathematics courses such as College Geometry, Problem Solving for Teachers, and History of Mathematics. \u00a0An example of a homework assignment I give based on the Riemann Hypothesis problem can be found at this link<\/a>. \u00a0When I use these problems for in-class work, I will typically pose the problem to the students without telling them it is unsolved, and then reveal the full truth after they have been working for fifteen minutes or so. \u00a0By doing this, the students get to experience the shift in perspective that comes when what appears to be a simple problem in arithmetic suddenly becomes a near-impossibility.<\/p>\n

Without fail, my undergraduate students, most of whom are majors in math, math education, engineering, or one of the natural sciences, are surprised that they can understand the statement of an unsolved math problem. \u00a0Most of them are also shocked that problems as seemingly simple as the Collatz Conjecture or the Erd\u0151s-Strauss Conjecture are unsolved — the ideas involved in the statements of these problems are at an elementary-school level!<\/p>\n

I have found that having students work on unsolved problems gets\u00a0them engaged in three ways that are otherwise very difficult to obtain.<\/p>\n

    \n
  1. Students are forced to depart from the “answer-getting” mentality of mathematics.<\/i> \u00a0In my experience, (most) students in K-12 and postsecondary mathematics courses believe that all math problems have known answers, and that teachers can find the answer to every problem. \u00a0As long as students believe this story, it is hard to motivate them to develop quality mathematical practices, as opposed to doing the minimum necessary to get the “right answer” sufficiently often. \u00a0However, if they are asked to work on an unsolved problem, knowing that it is unsolved, then students are forced to find other ways to define success in their mathematical work. \u00a0While getting buy-in on this idea is occasionally an issue, most of the time the students are immediately interested in the idea of an unsolved problem, especially a simply-stated one. \u00a0The discussion of how to define success in mathematical investigation usually prompts quality discussions in class about the authentic nature of mathematical work; students often haven\u2019t reflected on the fact that professional mathematicians and scientists spend most of their time thinking about how to solve problems that no one knows how to solve.<\/li>\n<\/ol>\n
      \n
    1. Students are forced to redefine success in learning as making sense and increasing depth of understanding<\/i>. \u00a0The first of the mathematical practice standards in the Common Core<\/a>, which have been discussed in previous blog posts by the author<\/a> and by Elise Lockwood and Eric Weber<\/a>, is that students should make sense of problems and persevere in solving them. \u00a0When faced with an unsolved problem, sense-making and perseverance must take center stage. \u00a0In courses heavily populated by preservice teachers, I\u2019ve used open problems as in-class group work in which students work on a problem and monitor which of the practice standards they are using. \u00a0Since neither the students nor I expect that they will solve the problem at hand, they are able to really relax and focus on the process of mathematical investigation, without feeling pressure to complete the problem. \u00a0One could even go so far as to evaluate student work on unsolved problems using the common core practice standards, though typically I evaluate such work based on maturity\u00a0of investigation and clarity of exposition.<\/li>\n<\/ol>\n
        \n
      1. Students are able to work in a context in which failure is completely normal.<\/i> \u00a0In my experience, undergraduates majoring in the mathematical sciences typically carry a large amount of guilt and self-doubt regarding their perceived mathematical failures, whether or not it is justified. \u00a0From data collected by the recent MAA Calculus Study<\/a>, it appears that this is particularly harmful for women studying mathematics<\/a>. \u00a0Because working on unsolved problems forces success to be redefined, it also provides an opportunity to discuss the definition of failure, and the pervasive normality of small mistakes in the day-to-day lives of mathematicians and scientists. \u00a0I usually\u00a0combine work on unsolved problems with reading assignments and classroom discussions regarding\u00a0developments in educational and social psychology, such as Carol Dweck\u2019s work on mindset<\/a>, to help students develop a more reasonable set of expectations for their mathematical process.<\/li>\n<\/ol>\n

        One of the most interesting aspects of using unsolved problems in my classes has been to see how my students respond. \u00a0I typically ask students to write a three-page reflective essay about their experience with the homework in the course, and almost all of the students talk about working on the open problems. \u00a0Some of them describe feelings of relief and joy to have the opportunity to be as creative as they wish on a problem with no expectation of finding the right answer, while others describe feelings of frustration and immediate defeat in the face of a hopeless task. \u00a0Either way, many students tell me that working on an unsolved problem is one of the noteworthy moments in the course. \u00a0For this reason, as much as I enjoy witnessing mathematics develop and progress, I hope that some of my favorite problems remain tantalizingly unsolved for many years to come.<\/p>\n

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        By Benjamin Braun, Editor-in-Chief, University of Kentucky One of my favorite assignments for students in undergraduate mathematics courses is to have them work on unsolved math problems. \u00a0An unsolved math problem, also known to mathematicians as an \u201copen\u201d problem, is … Continue reading →<\/span><\/a><\/p>\n

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