By Dimitrios Roxanas

A few years ago, when I started my tenured job at the University of Sheffield, one of my first initiatives was to start a problem solving seminar for students (undergraduate and graduate) and also academic staff. I have to admit I was a bit apprehensive when I pitched the idea of setting up these sessions (full disclosure: and for several months after!): I was a new and untested hire, with no obvious relevant credentials (I never competed in mathematical competitions beyond the local level), and a very distinct lack of seniority about me. Despite that, I found a lot of support from my more senior colleagues, some of whom are experienced recreational mathematics problem solvers: not only are they attending our sessions frequently, but they have also let me decide on the objectives that I wanted to pursue, and let me design the activities that I feel contribute to these goals.

Problem solving is certainly an integral part of learning and creating new mathematics, but for many mathematicians recreational problem solving is also a favourite pastime. I am mostly referring to solving problems from mathematical competitions -at the pre-university (e.g., International Mathematical Olympiad (IMO)) level, as well as the undergraduate level (Putnam, International Mathematics Competition for University Student (IMC)). But I also include problems from journals like the American Mathematical Monthly, Math Horizons, College Mathematics Journal and Mathematics Magazine. I have found myself drawn to the challenge but also to the artistic elements of problem solving, and I have come to appreciate its benefits to my research and to the way I absorb new mathematical ideas and I approach problems.

For my PhD I studied at the University of British Columbia (UBC, Vancouver, Canada) where Greg Martin and Dragos Ghioca were organizing a Putnam seminar, which was (is) in part training for the Putnam competition each December. The series was particularly popular with the more ambitious students, whether they planned to take the Putnam or not, and I could see why. No, it wasn’t just because of the free pizza and pop that the department was offering! The problems that were selected for the sessions were (very) difficult and yet deceptively easy to state. To solve them one need not have extensive background beyond the 1st or 2nd year of undergraduate studies, and still they would present an exciting challenge. You had to draw from rather basic material, but to make progress on those problems you would have to use this elementary material in creative ways. I always found this challenge very rewarding, and at UBC it was clear that I wasn’t the only one who felt this way.

My current problem solving sessions (PSS) are held every two weeks in both semesters. The sessions last 1.5-2 hours. They take place in a big lecture hall (when in person) or Blackboard–our Virtual Learning Environment—while all teaching and learning activities were moved online due to the pandemic. When in person, there is always pizza (which is a bit counterintuitive, I am always feeling “slower” after a couple of slices!) and very little structure. I distribute the handouts and then people move around, use the boards, chat, brainstorm, sigh loudly in frustration, but also exclaim in delight when a breakthrough comes. I love it! There are 110+ members in my mailing list for the group and we typically see 30-70 students and faculty joining the sessions. I estimate the regular members around 40, the vast majority of whom are undergraduate students.

Academic staff and graduate students are always welcome and especially faculty members. They attend frequently. Sometimes we even have 6-7 academics in attendance. I cannot emphasize enough how important I find the presence of faculty in these sessions. This is an excellent opportunity to increase faculty-student contact and make the students feel part of the academic community: academic citizenship extends to them too. It is a relaxed and friendly atmosphere, and it is much easier for the students to approach the faculty, ask for advice, bounce ideas off them, ask for their work to be checked, or just chat about academic life and their research.

From a pedagogical perspective there are even more benefits from this: I don’t share the problems and the solutions with the faculty members ahead of the sessions. On purpose. The idea to try this was a result of personal reflection. For many years I have been thinking about how to improve my own problem solving skills and become a better researcher, and I have frequently found myself frustrated reading solutions to problems and proofs in books and papers with no clue about the motivation that led to them. How did they know that such-and-such lemma would be useful? Did they know of this result in advance? If yes, how did they realize that this was applicable in this setting? How did they even think that this was the correct statement to prove? How can *I* learn to see what they see?

And of course, in my mind, “they” was usually a faceless authority; someone with such inherent mathematical talent, and lighting-speed intuition, that I could never hope to match. It wasn’t until I started working with Stephen Gustafson for my PhD that I realized that if those, very experienced and technically-skilled mathematicians, cared to share their insights and way of approaching problems (as he did), I could also start thinking like them. At first, you perhaps start by blindly applying to your own work their principles and rules-of-thumb, even if you haven’t fully internalized them or appreciated their significance in your context. Practice, and regular reflection, will eventually help you master it.

This is why I want students to see experienced mathematicians dealing with challenging problems, stripped of most context, and witness their thinking process. The problems I set are usually challenging enough to stump professional mathematicians. I would be the first to admit that there have been problems I couldn’t solve and many others that required a lot of time, effort and lateral thinking on my part before they budged. I want the PSS to also serve as training for research. And in research, we often don’t know how to get started or what tools can be useful. So what can one do? You can try small cases, or you can experiment by changing the problem a bit, e.g., by dropping or adding assumptions, working with “better” functions than you are “allowed to”, or maybe you can expand the space of admissible solutions. And I have found that one should always stay on the lookout for patterns. Why? Because when I spot a pattern or I make some observations, it usually leads (naturally) to formulating claims and conjectures. But that’s great news! This is something tangible I can try to prove (or disprove). It is unlikely (at least for me) that one will simply look at the problem and know immediately how to go about it.

All this is very different from what the students are used to from homework and exams. The context is always clear there (it must be relevant to the current chapter we are covering in class…), and there are a handful of possible tools that might be relevant when you are doing an exercise. To quote P. Zeitz (“Art and Craft of Problem Solving”, 2007, John Wiley & Sons) exercises *“can be easy or hard but never puzzling”*. There is obviously a clear place for exercises in the mathematical curriculum: this is how students learn the mechanics, and how they internalize the theory. Sure, I routinely use “mechanical” exercises to help them solidify their grasp of the material and appreciate the definitions, but also I expose them to harder ones, which I might scaffold if I deem it necessary. But these are still not problems. Problems are perhaps open-ended, and definitely not questions that one immediately sees how to approach. To be more specific about the type of questions I am talking about, consider the following (Putnam 2014, B1):

A *base 10 over-expansion* of a positive integer is an expression of the form

$N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0,$

with $d_k \neq 0$ and $d_i \in \{0,1,2,\dots,10\}$ for all $i$.

For instance, the integer N=10 has two base 10 over-expansions: 10=10⋅100

and the usual base 10 expansion 10=1⋅101+0⋅100.

Which positive integers have a unique base 10 over-expansion?

The hard part in this problem is to come up with the right claim to try to prove. Then actually proving that claim is a matter of technique and good bookkeeping. If you haven’t solved the problem, and I tell you that I claim (and then go about proving it) that the answer is “all the integers with no 0’s in their usual base-10 expansion”, you have every right to feel that the claim came out of the sky, and perhaps question your ability to ever do good mathematics. This is a prime example of a problem that invites investigation (and exactly why I picked it).

It’s impractical to share here all the experimentation we did with the students to approach this, so let me just outline some of the highlights:

Start by playing with several numbers, perhaps even randomly at first. “Playing” here refers to trying to express positive integers (with a varying number of digits) in two different base 10 over-expansions.

For example, one can start by looking at multiples of 10 and easily see that they can’t have a unique over-expansion. What about numbers smaller than 10? What about numbers between multiples of 10?

Can we be more systematic in our search? Maybe, let’s look at 2-digit numbers first, then 3-digit numbers, and so on?

List your findings. (we are trying to spot a pattern!)

So, for integers in the range (“ * “ is a placeholder for any digit 1-9) :

1-9: unique over-expansion

10-99: not unique if multiple of 10, i.e., of the form * 0. Hmm…

100-999: not unique if multiple of 10, i.e., of the form * * 0 , but wait! I can also find another representation for numbers of the form * 0 * . Hmm…

1000-9099: let’s see, I can find a different to the standard base-10 representation for numbers of the form * * * 0 , * * 0 * , * 0 * * . Hmmm…!

This is already suggesting a pattern! All these have a 0 in their decimal expansion; am I looking for all the integers with no 0’s in their usual base-10 expansion then?

Once a concrete, “mathematical” claim has been established, a few went about proving it (which isn’t too hard, I invite everyone to try). This is an example of a problem where if you spend your time staring at it, without getting your hands “dirty” with some examples, you will likely never even figure out how to get started!

I also try to remind students that sometimes you get fixated on the wrong pattern, or you didn’t check enough cases for an educated conjecture and what you are trying to prove is actually not true in general. And you know what? That’s fine. We still learn from approaches that don’t work (and they can inform successful ones). Also, many famous mathematicians claimed statements that ended up being false. We are in good company!

Seeing the faculty members approach these problems the same way they would approach a research question has been instructive and motivating for my students. Interestingly, since we started this activity at Sheffield a few years ago, there has been a dramatic increase in the uptake of undergraduate summer research projects. As the coordinator of both programs I have seen a significant overlap between those who attend the PSS and those who pursue summer research. In some cases, undergraduate students started attending research seminars, or formed advanced study groups and in one case they even started working jointly on an open problem!

I have found that this approach to problem solving (i.e., investigation and experimentation), or rather, this attitude, is not necessarily a developed instinct in our students, not even in the better ones. But, again agreeing with P. Zeitz, it can be *taught and nurtured.* Encouraging in-session interaction with faculty is one way to do this, but this is just one component of what I try to do. I make additional progress to this goal also by how I present my solutions to these problems, and of course by my choice of problems. Clearly the two are connected: appropriate problems will give me the chance to highlight the steps in this approach.

In writing solutions to such problems, my goal is to shed light on the thinking process, not to present a linear exposition as you would see in a textbook. So I don’t hesitate to take a nonlinear path, sharing my thoughts along the way, showing dead-ends, adding remarks that explain the state of my insight into the problem, but also how my intuition is strengthened and my insight deepened as I experiment, calculate, and develop small-scale plans. And again, isn’t this how we do research? In my opinion that’s not only an appropriate approach but also a desirable one. To quote Steven Krantz (from “How to Teach Mathematics”, 1999, AMS)* “…we learn inductively (from the specific to the general) rather than deductively (from the general to the specific). The deductive model is highly appropriate for recording mathematics, but it does not work for discovering mathematics…”. *

* *As an educator, it delights me that this activity allows me to aim to go beyond the low *(knowledge, comprehension, and application)* stages of expertise (in Bloom’s taxonomy) to those of *analysis and synthesis*. And this is exactly because these problems require investigation, speculation and conjecture before one can come close to a solution. It is exactly the element of investigation that is present in analysis and synthesis that students rarely get to experience when they learn mathematics.

Pedagogical reasons aside, the impact of the PSS on engagement have been clear at Sheffield. The sense of community has been much more pronounced. Student-faculty interaction has increased. This has been seen very positively by the students: many are starting to feel that they are members of the department’s academic community. Previously-isolated students have found like-minded peers and they are now studying together, going beyond the curriculum, and together engaging in research. !

Having said that, there is still a lot to be done. The students who participate in these activities are admittedly among the higher-performing ones, the more likely to engage if we offer them the opportunity, and they are usually the ones who are more inclined towards Pure Mathematics. How to reach out to more students? My first attempt was to include a second category of problems, or rather two categories. One is basically challenging brainteasers in the form of technical interview questions. The other is “algorithmic puzzles”, challenging problems that can be approached in a systematic way (perhaps after some recasting into a more amenable form), for example using Dynamic Programming. My hope was that this would appeal to students who are more interested in programming or something more “applied”, and to also offer some training for those challenging quant/software engineering/data science interviews. My next project will be to introduce a second series revolving around mathematical modelling, which hopefully will allow us access to a different subset of the student population.

One of the hardest aspects in organizing this activity was striking a good balance: and trying to be more inclusive, trying to adjust the level of the problems so that students don’t get totally discouraged, but at the same time making them hard enough for students to think outside the box, embrace new techniques and the “investigation-first” attitude that I am hoping to promote; and to also build resilience in the face of challenge. I don’t claim to have found an algorithm for selecting good problems, and I am sure some keen students have been occasionally discouraged despite my best intentions. Other than managing expectations and working towards a supportive environment where “failing” is seen as a necessary (and expected) step of the process, I have tried to adapt too: I have been starting the sessions off with quick, “warm-up” problems, still not trivial, that everyone has a chance to solve. (Groups of) students work independently, so nobody will notice if you spend the whole session working on the “warm-up problem”. Note to self: stop calling it a “warm-up” problem?

Another source of “anxiety” for me comes from the workload involved. In the past 3 years, I have typed more than 300 pages for my Problem Solving sessions. I have compiled “theory” (methods of proof, specific problem solving strategies and tricks) from various sources, I have created about 40 handouts, and I have written very detailed solutions to every problem. What does that entail? First, I identify the topics I want to discuss and look into my sources for good ways of motivating a method and also examples. I always solve the problems myself before looking at the official solutions which means that I can get (very) stuck, and I frequently have to discard problems (maybe because there was a very specific trick, or you needed to know a rather esoteric result). Then I type up everything. Looking back I usually need 10-15 hours of preparation for a single problem solving session, and so far I haven’t been able to recycle work from previous years — I think that I will eventually try for a 4-year cycle and hope that my colleagues won’t mind seeing some of the same problems every 4 years!

I want to quickly discuss the transition to online sessions, a necessity that the Covid-19 pandemic brought on. This really required a new approach. You can’t possibly replicate the big room-with-pizza-and-people-moving-around ambience in an online session, and I was well – aware that no matter what we tried, the social aspect would suffer somewhat. One of the challenges in online sessions is that only one person can be speaking at the same time (unless of course you are using “breakout rooms”). Another challenge is that it would be difficult to ensure intervals of peace and quiet for some serious thinking to take place. I realized that I had to avoid long silences, which would be rather awkward, so I changed the format of the sessions. Instead of sharing the problems for the first time at the session, I would send them 3-7 days in advance, and ask people to volunteer to present their solutions (or just things they tried). They had until the night before the session to let everyone know if they wanted to do that (we always had at least one volunteer, either student or staff). But that was just part of it, because that didn’t guarantee we would always have fun in the sessions. And why do it if it’s not going to be fun?

The other feature that I introduced in the online versions was “quick fire problems”. By that I mean a selection of interesting problems (typically from the UKMT Senior Mathematical Challenge, the AMC or the AIME) that can be tackled in 5-10 minutes. We would start the session with a couple of those, and then intersperse them among the more “serious” problems’ presentations and discussions. So I would give 5-15 minutes, depending on the difficulty of the problem, which was protected quiet time during which students were encouraged to send me private messages with questions/solutions. Once the time was up, I would ask for volunteers to present their solutions, typically using interactive online whiteboards (but also visualizers). This activity generated quite a bit of interaction, with the additional benefit of creating opportunities for more students to feel successful with solving less (but still) challenging problems.

After 19 months of social distancing and multiple lockdowns, UK universities have recently returned to face-to-face teaching. And what better way to celebrate the return to in-person activities than by attacking hard mathematical problems (in our properly-ventilated lecture theatres)? Our problem solving sessions this term have hit an all-time-high attendance record: 65 students and 5 staff in one session, and we are regularly hosting 30-40 students. As you can imagine we’ve been having a blast! We can’t wait to work on the problems from this December’s Putnam Competition.

I want to conclude by thanking my colleagues at the University of Sheffield, especially (in alphabetical order) James Cranch, Neil Dummigan, Frazer Jarvis, Jayanta Manoharmayum, Fionntan Roukema, Evgeny Shinder, and Alan Zinober, fok r supporting these activities, for their input and feedback, and more importantly for contributing to a better student experience.

**About the author:** Dimitrios Roxanas is a mathematician at the University of Sheffield, UK. He earned his PhD at the University of British Columbia and then moved to the UK, spending a few years at the University of Edinburgh and the University of Bath before moving to Sheffield. At the University of Sheffield he is the organizer of a problem solving seminar and the coordinator of under