In honor of the strong performance by Williams College on the Putnam Math Competition (placing 14th out of over 500 teams, with two students in the top 200 and two in the top 500 in North America), this is a good time to talk about why these competitions are worth doing. There are many reasons for doing mathematics, ranging from applications to real world problems to just loving a challenge. Often math competition problems fall in the latter category, though the truly good questions (in my opinion) have connections to other parts of mathematics or allied disciplines. Here are a few of my favorite contest problems from the years, and some of the applications:
(1) Let N be a large number; how should we write N as such that the product is as large as possible and each ai is a positive integer (or, more generally, each ai is just a positive number)? Note is arbitrary — we get to choose it as well as each . This has applications to building the most efficient computer for data storage.
(2) Consider 10 identical cookies and 5 distinct undergraduates — how many ways can we divide the cookies among the students where all we care about is how many cookies a student gets? While this problem can be solved by brute force, the elegant approach is just one line, and is the beginning of a whole set of problems in additive number theory.
(3) Can one place 5 queens on a 5×5 chess board such that there are three squares they do not attack (and thus we may safely place three pawns on the board)? Again, while this problem can be solved by brute force, there is a more elegant approach which is but one of many instances of a powerful technique in mathematics: the duality principle (also known as: instead of solving your problem, I claim I just need to solve a related, but simpler problem). One of the biggest applications of this is in Linear Programming.
(4) Imagine Williams is a nuclear power and whenever any three professors agree, they can fire the warheads; however, no two professors ever suffice to launch our weapons. To ensure this is the case, someone suggests the possibility of having a three number password, say (a,b,c), to launch. A little thought indicates a potential problem. One cannot just tell each general one of a, b or c (as then it is possible that some subset of three generals won’t know a, b and c); however, if a general knows two of (a,b,c), then a set of two generals can launch the missiles! What information should be given to the generals so that any three can find (a,b,c) but no two can? This is but one of many examples of cryptography, whose applications are well known to us all.
If you have any thoughts about these problems (or others you find interesting), please post comments below. If you’d like more problems, check out my riddles page. If you google “math riddles” it’s usually in the top 10. It’s been a lot of fun maintaining this page. I’ve been contacted by teachers all over the world who use these and similar riddles in their classes as ways to excite students about math and its applications. You should also check out the monthly math conundrum (click here for current and past problems).
The Putnam is just one of many math competitions, though this is the most famous. Each year we also compete with Middlebury in the Green Chicken contest (winner gets the porcelin Green Chicken; sadly Middlebury won this year and has it, but if you want to see what it looks like come to a colloquium, as its sibling the Brown Chicken brings cookies afterwards).
For at least the past few years, a couple of times a week some students and professors here gather to discuss interesting problems and their applications and connections. Currently we’re meeting in Paresky 207 at 5:45pm on Mondays when there isn’t a math/stats dinner, and Tuesdays at 5:45pm at Mission in the Dennett Private Dining room; for more information see our homepage, and click here for a summary of some common problem solving techniques. I hope to see you around!