Computers on the Qualifying Quiz

There are some problems in mathematics that no one has yet been able to solve without a computer. Our quiz problems are not like that! With each of them, once you find the right way of thinking about it, a computer becomes unnecessary.

Of course, for those who like programming, a computer can be a good way to come up with ideas or to test hypotheses. However, the ultimate goal is to figure out the hidden pattern in each problem– to understand what is really going on.

Consider, for instance, Problem 2 from the Mathcamp 2005 qualifying quiz:

Eric and Matthew are visiting the town of Descartes, Quibec, and have managed to lose each other. Descartes has five equally spaced streets running east-west, cut by five equally spaced streets running north-south; each of the 16 blocks is a perfect square. Eric is at the southwest corner of town, and decides to look for Matthew by walking towards the northeast corner along a path of shortest length. At each intersection where he has a choice between north and east he chooses which way to go randomly, with both directions equally likely. Meanwhile, Matthew is at the northeast corner, deciding to look for Eric by walking to the southwest corner in exactly the same manner. Both start walking at the same time, and walk at the same speed. What's the probability that they meet?

If you were to write a program to generate all possible paths that Eric and Matthew could take and check in each case whether they meet, you would be missing the point of the problem completely. The point is to derive the probability from the general properties of such a situation.

This is not just an annoying restriction we have dreamed up to make your life more difficult. In real mathematics, it is often possible to check certain special cases (such as a five-by-five grid of streets) with the help of a computer, but the solution of a general problem (such as an n-by-n grid of streets, for arbitrary n) still practically always requires human insight – and in almost all cases, it is the solution to the general problem, rather than the specific one, that is more mathematically interesting.

So feel free to use the computer in the initial stages of your exploration. However, if your final solution still relies on a computer, you are almost certainly missing some key insight.