By Deirdre L. Smeltzer

Joseph Gallian and Carl Schildkraut are the inaugural co-editors of the Mugs to Donuts (MtD) e-newsletter, a recently-launched MAA publication aimed at connecting with current and future Putnam competition participants. I sat down with Joe and Carl to learn more about their interests and find out their plans for Mugs to Donuts. Many of you already know Joe through his extensive MAA and wider involvements: former MAA president, the organizer of over 40 summer REUs at the University of Minnesota Duluth, and the author of the popular Contemporary Abstract Algebra textbook, among other things. (You can learn more about Joe here.) Now let’s get to know Carl, a current MIT math major.

Getting to know Carl

How did you become interested in problem-solving?

For as long as I can remember, I’ve been interested in math. I participated in my first math competition (a small local contest) in third grade, and was immediately hooked. At that age, the questions were mainly just computations and word problems, but this activity led to me participating in more and more math competitions as I got older. It was through the American Mathematics Competitions (AMC) series a few years later, and particularly through the open-ended nature of the USA(J)MO [USA (Junior) Mathematical Olympiad], that my love of mathematical problem-solving really began.

You participated in the Putnam competition in 2019 as a first-year college student. Describe that experience.

It’s very odd to come out of a 6-hour exam and wish that it were longer.

To elaborate: the math olympiads I had experience within high school were usually three problems in 4.5 hours. In such an exam, unless your ideas are really time-consuming to attempt, it’s relatively difficult to run out of time before one runs out of ideas. Each session of the Putnam, on the other hand, is “only” 3 hours, and has twice as many problems — instead of 90 minutes per problem (on average), you get 30. It’s so easy to get sucked into an idea that looks promising and not realize that half your time is gone and you still haven’t read two of the problems. However, once the experience is over, it’s a great feeling to know you gave everything you had to something so daunting.

What has been your most satisfying problem competition experience, to date?

I’ll go with the 2020 USOMO [US Online Mathematical Olympiad]. This experience was especially satisfying because it was the first “major” competition that I got to see from the other side, and in which I was able to help out in all phases, from writing problems, to voting for problem selection and devising alternate solutions, to grading the submissions afterwards. It was a lot of work, but it’s exciting to see it come to fruition. Especially with the uncertainty about how math competitions could fit into a pandemic world, the 2020 USOMO made for a very satisfying experience.

What motivates you to give back to the mathematical community—by serving as co-editor of MtD, working as a Math Olympiad Program (MOP) TA, writing math contest problems, etc.?

A lot of it is because I feel so lucky for all the resources that I had, all the people who helped me along the way, and all the connections I made. The mathematical community has shaped, and continues to shape, my life and career, and it feels like somewhat of a duty to help to create that community for those a few years younger.

However, I’m afraid this makes things like serving as MtD co-editor, working as a MOP TA, and writing problems sound a lot more altruistic than they are for me—I wouldn’t be doing any of these things if I didn’t genuinely enjoy them! As a MOP TA, for example, I get to try a lot of enjoyable problems to set the tests, to stay connected with people who were instructors/TAs/students when I was at MOP, and to meet and get to know new students. As MtD co-editor, I’ve gotten to meet and work with interesting people and to put my random internet searching to good use finding worthwhile content for the newsletter. When writing math contest problems, I love grappling with complex ideas, trying to figure out how to turn them into elementary statements, and thinking about the contestant experience. I’m glad that these things have value to the math community that I care about so much, but, independently of that value, these are all things I enjoy doing.

What makes a Putnam problem appealing? (What is a “good” problem?)

In my mind there are lots of different ways in which a problem can be appealing. I’ll use a few examples from recent exams to illustrate this.

My favorite problem on this past year’s exam was B6, because my immediate reaction to it was “there’s no way this can possibly be true.” There’s a sequence of positive and negative ones that “should” be completely random, but yet there are always supposed to be more positive ones than negative ones no matter how far you count? Madness. There’s something very appealing about a problem statement that looks false, and turns out to be not only true but has a solution simple enough to put on a contest.

Another way a problem can be appealing is as a “puzzle”—for an example, Putnam 2019 A4 is a yes-or-no question (asking if a function with certain properties must be the zero function), without an obvious answer. To solve it, you really have to understand what’s happening and change your point of view back and forth from “let’s try to find an obstruction” to “how can I actually construct such a function.” While the statement isn’t as shocking as 2020 B6, it’s hard not to enjoy the solving process here, and even if you don’t solve it, you come out having thought about cool things.

The last type of problem that I find very appealing is the “‘shallow’ problem that hints at something deeper.” I’m going to cheat a bit by referencing a very recent IMO problem, 2021 IMO 2 and justify this inclusion by saying that it would fit much better on the Putnam than a high school contest. It’s frighteningly difficult, and has four distinct solution paths that I’m aware of. They’re all quite different, ranging from an impossibly tricky elementary solution, to a “simple” reduction of the (algebra) problem, to a question of combinatorics, to approaching it with linear algebra or calculus. The beautiful thing about this problem is that every approach involves a winding road through a strange landscape, very different from either the setting of the problem or its eventual resolution. However you solve it, you have to come up with or introduce new ideas or new types of math. Problems like this help to expand contestants’ mathematical horizons, which is something that all competitions should strive for.

Why do you think students should participate in Putnam competitions?

In my mind, the value of competitions like the Putnam is significantly less about the specific problems and much more about the sort of open-ended mathematical thinking that such problems require. In pretty much everything in life outside of school assignments/exams, challenges come with relatively little fixed context; when searching for a solution, one has to draw directly from one’s past experience, not “we just learned about X, so I’m probably supposed to use X.” The Putnam facilitates making these kinds of connections with prior knowledge or past experiences in a very time-constrained setting, and I think that’s an invaluable skill, in math and beyond.

Joe’s Connection to the Putnam

Joe, You’ve run an incredible number of REUs. How many of your REU students have been involved in the Putnam over the years?

Close to 100% of the 273 participants who have been in my REU since 1977. Forty-three were Putnam Fellows; 80, if you count multiplicity. Of the six people who have been four-time Putnam Fellows since my first undergraduate research program, four were in my REU. The Putnam has a wonderful history with many very accomplished participants, including Nobel Prize winners, Abel prize winners, Fields medalists, Presidents of the AMS, and members of National Academy of Sciences. The fact that I have worked with so many people who have done very well on the Putnam greatly adds to my interest in its history.

And, what motivates you to give back to the mathematical community?

Even as a young boy. I wanted to help others. I had my first leadership experience at about the age of 14 when my local YMCA athletic director asked me to help him with various Y activities. Just like serving MAA, I considered it an honor to contribute in some way. As an undergraduate at Slippery Rock, I led an effort to start a math club and a tutoring service. As a grad student at Notre Dame, I joined the South Bend Audubon Society, and a few years later, I was honored to be asked to serve as President. I never sought such positions but I did seek to serve in some capacity.

Launching Mugs to Donuts

Joe and Carl, what goals do you have for the Mugs to Donuts newsletter?

Joe: I will give you four goals: to make it informative, interesting, fun, and different for people who enjoy math competitions.

Carl: My main goal for the Mugs to Donuts newsletter is to showcase interesting things. One of the things that’s so beautiful about math is the number of different settings, both “big” (e.g. computer graphics, economics, pure mathematics) and “small” (e.g. random curiosities, mathematical puzzles), in which math plays an important role. Once you start thinking mathematically, math shows up in everything. A lot of publications (and in fact a lot of the content in Mugs to Donuts) focus on the big stuff—the career options, the “proper” research math, the things that make our world tick. But I’m hoping to showcase some of the really fascinating little things—places that math randomly shows up in popular culture, new and inventive puzzles and games, and the occasional mathematical curio—the things that make it really cool to be mathematically literate. I think the Putnam exam straddles the line between “big stuff” and “small stuff” in a very valuable way, and I’m hoping that we can strike an intriguing balance in this newsletter.

Deirdre Longacher Smeltzer is currently MAA’s Senior Director for Programs and formerly a math professor, department chair, and undergraduate dean. Working with and learning to know fascinating, talented people is one of the best parts of her job.