G4G is pleased to announce that Professor Manjul Bhargava of Princeton University will make a public presentation in Atlanta, on “Poetry, Drumming, and Mathematics” (2:30 pm Sunday, April 15, 2018, at Fulton County Central Library in downtown Atlanta, One Margaret Mitchell Square, 30303). This will include musical performance. Professor Bhargava is a winner of the Fields Medal as well as an accomplished tabla player who studied under Zakir Hussain.
He will also speak on “Some Magical Numbers and Their Uses in Magic and Mathematics” at the G4G13 conference.
We were delighted to chat with Professor Manjul Bhargava this week about the role that puzzles, the Rubik’s cube, and recreational mathematics (not to mention skipping classes!) has played in his life, and why he loves the natural connections of math to music and poetry.
1. What are your first memories of mathematics or recreational mathematics?
I always enjoyed mathematics, as far back as I can remember. I loved playing with numbers and shapes, and looked for them everywhere, including in nature, in floor tiles, in wallpaper… everywhere. My favorite toys tended to involve shapes, games, numbers. I loved puzzles, word games, number games, as well as poetry and music, all of which I considered to be very similar in nature.
The first nontrivial math problem that I ever solved as a child was sort of a puzzle that I made up on my own. I was around 7 or 8 years old. I used to really enjoy stacking oranges, meant for the family juicer, into triangular pyramids. The question naturally came up: how many oranges does one need in order to make a triangular pyramid that is $n$ oranges wide, i.e., having n oranges on each edge?
I loved this question, and remember thinking about it off and on for some time – perhaps weeks. It was such an exciting moment when I finally figured out the answer: n(n+1)(n+2)/6. Thus, for example, if one has four oranges on each side of the pyramid, one needs exactly 4(5)(6)/6 = 20 oranges! And I could actually take 20 oranges and make the pyramid! That moment was incredibly exhilarating: it showed me the power of mathematics to predict phenomena, and to explain patterns. And because I had come up with the answer on my own—and in a couple of different ways—it demonstrated to me the creative process of mathematics, and that there is no one right way of approaching a problem. That was a truly amazing lesson for me to learn early on—a lesson that unfortunately does not often come up in mathematics classes, where generally one is just told the exact fixed steps one must take in order to solve each problem.
2. So it seems that you had your early introductions to mathematics through recreational mathematics, on your own, rather than through mathematics classes?
3. What was your mathematical journey after that? What role did recreational mathematics continue to play in your mathematical development?
Growing up, I always knew that I loved mathematics but that, alas, I didn’t like mathematics classes very much—and, as a consequence, I didn’t attend very often. My family was remarkably cool about it despite complaints from school. Sometimes I’d even take off months at a time from school, and spend it in India with my grandparents, where I’d learn tabla from my teacher there and also Sanskrit poetry from my grandfather—and I saw so much mathematics in each.
My mother is a mathematician, so of course I’d always learn fun mathematical tidbits from her growing up. When I asked her questions, she’d always encourage me to figure them out on my own (like the oranges question!), sometimes with gentle hints. Sometimes I’d skip school and attend her classes at Hofstra University, which I enjoyed more than going to mathematics class in school.
Working on solving Rubik’s cube with my mother as a child remains one of my favorite memories from childhood. That’s why I am really excited to be meeting the great Erno Rubik at this year’s G4G—I have been a huge fan since childhood! I remember taking my time enjoying all of Rubik’s great puzzle inventions that I could find, including Rubik’s Clock and Rubik’s Magic.
Eventually, when I was around 12 years old, through my puzzle explorations I of course also had the good fortune of discovering the works of Martin Gardner. They inspired me a huge amount, and gave me something far more enjoyable to do than go to math class! I also read other recreational mathematics and puzzle books, such as those of Raymond Smullyan, and all of these works definitely had a great influence on me as a playing and playful mathematician.
I still read Martin Gardner’s works to this day, as well as other works of recreational mathematics, and am always playing around with mathemagic in some way or other. Recreational mathematics often leads to serious research mathematics as
well, and it certainly has for me. I hope to talk about some personal
experiences of this kind in Atlanta.
Years later, when I think about this, and my experiences with recreational mathematics, I wonder: if doing puzzles and games, thinking about poetry and music and magic, and reading Martin Gardner and Raymond Smullyan, etc., were so inspirational to me (and to so many others) in learning mathematics and becoming a mathematician, why should such ways of learning mathematics not be an integral part of mathematics class in school? It baffles me that they are not!
Obviously, I personally think that such fun, puzzle-based activities should be incorporated at every level of mathematics teaching. I also do my best to incorporate such activities in my own teaching whenever possible, and that has certainly been a very successful and important part of teaching for me.
Martin Gardner was of course also a firm believer that recreational mathematics should be incorporated into the regular mathematics curriculum, and I couldn’t agree more.
4. We understand you also now teach a popular freshman seminar on recreational mathematics – is that right? Can you tell us a little bit about it?
Yes, for the past few years I’ve been teaching a freshman seminar called “The Mathematics of Magic Tricks and Games”; it’s been so much fun for the students and of course for me! Every class starts with a magic trick or the rules of some game (with a couple of demo runs so that everyone sees how the game works), and then the goal of the rest of class is to understand and explain what happened in the magic trick, or how to win at the game! Every game or trick has some key mathematical principle at its core, so by the end of class everyone discovers this principle through discussion and playing around and gentle hints. The homework for the week then is to become proficient at performing the trick or playing the game, which in essence means that they become proficient also in the mathematics behind the trick/game. And they never forget it, even after I meet them after years—because it’s very hard to forget a principle or theorem in mathematics when you’ve discovered it yourself, and that too in a fun way.
Some of the principles that come up in talking about such mathemagical mathematics turn out to be related to major unsolved problems and research work going on around the world in pure and applied mathematics (sometimes even my own research); when that happens, I take the time out to talk about that as well, which helps to connect all the serious fun and games to deep and important mathematical problems that are the focus of major research.
5. We hear you also incorporate music and poetry in your classes. Can you tell us a little bit about that as well?
Because I learned much mathematics through music and poetry, I do like to offer classes that incorporate these elements as well: for example, frequencies in music that sound pleasant to the ear are those that “resonate” with each other, i.e., the ratios of the frequencies being used should be simple whole number ratios for music to sound good. Given that fact, what scales of notes should optimally be used in music? This question is a beautiful mathematical problem in art, to which different cultures around the world found different and beautiful solutions.
Rhythm, and in particular, the rhythms and meters of poetry and music, is another area where there is much incredible mathematics. I hope to talk about a number of these aspects while in Atlanta! Very much looking forward to it, and to learning and enjoying a lot!!
6. We look forward to having you too!
Thanks so much—excited to be coming to G4G after wanting to for so long!!