# Mathematics

This is an index of writing on mathematics.

At some point, I hope to put up my lecture notes for a Splash[1] class I taught in 2009 called “The Joy of Eigenvalues”:

Linear recurrence relations, generating functions, graphs, oh my! We will explore the connection between these seemingly unrelated ideas by the numbers which are intrinsic to their structure. Applications include Google PageRank, vibrating systems, and population dynamics. We will start with the definition of an eigenvalue and continue from there (with many examples and λ’s).

## 1. Small problems

These are just some small problems with their solutions.

This one was given to me in my 5th grade math class.

Suppose you have an array of lockers, labeled 1, 2, 3, and so on, and they’re all closed. By “toggling a locker,” we mean opening it if its closed and closing it if it is open. You have a lot of time on your hands, so you decide to do the following. First you toggle every locker, then you toggle lockers 2, 4, 6, and so on (multiples of two). Then you toggle lockers which are multiples of three, and then you keep doing this process of toggling the multiples of four, five, six, and so on. Which lockers are open by the end, if any?

You can try this experimentally, and with about 20 lockers you can see the pattern: the lockers which are open are those which are squares! (i.e., 1, 4, 9, 16, …).

Why is this? First, to determine whether a particular locker is open or not, all you need to know is whether the number of times it was toggled is even or odd. Each locker is toggled once for each of its divisors, and divisors usually come in pairs (if d is a divisor of n, then n/d is a divisor of n, too). But are d and n/d distinct? If n is a square, then d being its square root gives d = n/d. If n is not a square, then d and n/d are always different. This means that a square has an odd number of divisors, and a non-square has an even number of divisors; hence, the squares are the ones with the open lockers!

[1] This is a program put on by ESP at MIT every year where hundreds of high-school (and younger) students come for a weekend to be taught about virtually any topic.