*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. An unsolved math problem, also known to mathematicians as an “open” problem, is a problem that no one on earth knows how to solve. My favorite unsolved problems for students are simply stated ones that can be easily understood. In this post, I’ll share three such problems that I have used in my classes and discuss their impact on my students.

**Unsolved Problems**

*The Collatz Conjecture*. Given a positive integer \(n\), if it is odd then calculate \(3n+1\). If it is even, calculate \(n/2\). Repeat 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. If 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. The unsolved question about this process is: If you start from any positive integer, does this process always end by cycling through \(1,4,2,1,4,2,1,\ldots\)? Mathematicians 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.

*The Erd*ő*s-Strauss Conjecture*. 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? For example, for \(n=3\), we can write \[\frac{4}{3}=\frac{1}{1}+\frac{1}{6}+\frac{1}{6} \, . \] For \(n=5\), we can write \[ \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20} \] or \[\frac{4}{5}=\frac{1}{2}+\frac{1}{5}+\frac{1}{10} \, . \] In 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 \(c\)? Again, most mathematicians believe that the answer to this question is yes, but a proof remains elusive. This question was first asked by Paul Erdős and Ernst Strauss in 1948, hence its name, and mathematicians have been working hard on it ever since.

*Lagarias’s Elementary Version of the Riemann Hypothesis*. For a positive integer \(n\), let \(\sigma(n)\) denote the sum of the positive integers that divide \(n\). For example, \(\sigma(4)=1+2+4=7\), and \(\sigma(6)=1+2+3+6=12\). Let \(H_n\) 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 problem is equivalent to the Riemann Hypothesis, a famous question about the complex roots of the Riemann zeta function. Because it is equivalent to the Riemann Hypothesis, if you successfully answer it, then the Clay Mathematics Foundation will reward you with $1,000,000. While the statement of this problem is more complicated than the previous two, it doesn’t involve anything beyond natural logs and exponentials at a precalculus level.

**Impact on Students**

I’ve used all three of these problems, along with various others, 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. An example of a homework assignment I give based on the Riemann Hypothesis problem can be found at this link. When 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. By 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.

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. Most of them are also shocked that problems as seemingly simple as the Collatz Conjecture or the Erdős-Strauss Conjecture are unsolved — the ideas involved in the statements of these problems are at an elementary-school level!

I have found that having students work on unsolved problems gets them engaged in three ways that are otherwise very difficult to obtain.

*Students are forced to depart from the “answer-getting” mentality of mathematics.*In 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. As 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. However, 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. While 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. The 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’t 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.

*Students are forced to redefine success in learning as making sense and increasing depth of understanding*. The first of the mathematical practice standards in the Common Core, which have been discussed in previous blog posts by the author and by Elise Lockwood and Eric Weber, is that students should make sense of problems and persevere in solving them. When faced with an unsolved problem, sense-making and perseverance must take center stage. In courses heavily populated by preservice teachers, I’ve 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. Since 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. One 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 of investigation and clarity of exposition.

*Students are able to work in a context in which failure is completely normal.*In 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. From data collected by the recent MAA Calculus Study, it appears that this is particularly harmful for women studying mathematics. Because 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. I usually combine work on unsolved problems with reading assignments and classroom discussions regarding developments in educational and social psychology, such as Carol Dweck’s work on mindset, to help students develop a more reasonable set of expectations for their mathematical process.

One of the most interesting aspects of using unsolved problems in my classes has been to see how my students respond. I 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. Some 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. Either way, many students tell me that working on an unsolved problem is one of the noteworthy moments in the course. For 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.

Thanks for the great ideas Ben!

I currently have some students working on Sylver Coinage as part of a final project in a Number Theory class. They have come up with a few strategies on their own and are intrigued by the fact that more strategies haven’t been discovered. It’s even more interesting that there are nonconstructive proofs for strategies. Because these students know the problem is unsolved, they are more apt to be proud of small steps forward.

I’d love to hear more ideas of what is done in other classes.

I hadn’t heard of that problem before, it sounds like an awesome project!

Came here from https://news.ycombinator.com/item?id=9537802.

In the Erdos-Strauss conjecture why is it 4/n. Clearly one is limited by the highest number achievable with the fractional terms on the right, but why not the general question x/n = Sum 1/a[i] for i from 1 to (x -1)?

[attempted to render that in mathml:]

xn=∑i=2i=x-11ai

Has a form of this been solved generally for other x values ( > 3) and 4/n is an outlier??

In my mind geometric solutions for the first two are so obvious (!) however in my current state of mind I’m far more inclined to the pragmatic “well you can show if it’s true for any particular n you care about, close enough” – think I’m losing my youthful inquisitiveness!

Some work has been done on the more general case, see Terry Tao’s comments on his blog https://terrytao.wordpress.com/tag/erdos-straus-conjecture/ and the references therein.

This is fantastic and inspiring! Thank you.

Hey, Beth, thanks for mentioning the Coinage Problem. I have been teaching Discrete Math (including some number theory) at the University of Alabama for years and one of the problems in our book has to do with making change. Once I read your note, I googled (or do I mean capitalized Googled?) the coinage problem and, lo and behold, it has a long history! Sylvester! Frobenius! Wow…I am going to try this question in my class this semester to see if the kids can guess and then prove Sylvester’s (a-1)(b-1)-1 theorem! I don’t think I’ll ask them to prove the Lagarias version of the Riemann Conjecture, though…

I wish that 30 years ago I had studied under you. At first read, I thought this was cruel and unusual punishment, then I realized your method.

Wonderful… you are someone who should keep teaching forever. Please keep doing what you’re doing.

If I didn’t live so far from Kentucky I would be begging to audit one of your classes.

Hi Ben,

I am not a mathematician but I have a question about the The Collatz Conjecture. If we take any positive number and lets say it is an even number then, we will divide it by 2. and obviously we will get the Even number and then again we will divide it by 2 and again we will get the even number and this process will continue until we come to 4 and then we will divide the 4 by 2 and we will get 2 and then divide 2 by 2 and we will get 1 so it will always end with 4.2.1. Incase of odd number we will do the 3n+1 that means we will get the Even number and then again the Even number process that means we will also end up with 4.2.1. So what is need to be proven here. It is obvious and can be proved with the simple number theory.

Your suggestion only works for powers of 2. If you have a number like 10, which is even but has odd factors, then it isn’t clear what will happen.

Wonderful post! I will be sharing this in my on-going conversation about authentic curriculum in elementary mathematics. I use open problems frequently in my 4th and 5th grade maths classes, especially during, but not limited to, our units on number theory. The students love the work. It is fun, engaging, motivating, empowering and capitalizes on their natural curiosity. Many students stick with the problems after we’ve moved on because they hope one day someone will find the proof and move them beyond conjecture. Like Julie Robinson and the diophantines, it didn’t have to be her, she just hoped for a proof in her lifetime so she could celebrate it. Furthermore, these problems do a wonderful job of showing students that mathematics is not dead, but full of discovery and humanity. They are motivated to learn more about the players and their stories.

Hi Ben,

I don’t want to waste you time but I really find it fascinating. So I Just wanted to understand it more. If I take 10, its a even number and I divide it by 2, I will get 5 and then (3x+1) so I will get 16 and then divide it by 2, will get 8 and then 4,2,1. So whats the problem in it. Just to understand this better.

Thank you so much for such interesting post.

This page: https://en.wikipedia.org/wiki/Collatz_conjecture has lots of good information about the details of the conjecture. At the end of the page, there is a nice list of references where you can find more detailed discussions about why the problem is difficult.