New York city recently opened the Museum of Mathematics, the United States’ first museum dedicated exclusively to mathematics.  It opened on December 15 (my birthday!) and it’s tons of fun.  If “museum of mathematics” conjures the image in your mind of a framed painting of the quadratic formula or the bust of a dead Greek intellectual, you would be very pleasantly surprised.  The museum is full of puzzles, games, toys, computer simulations, and innovative hands-on exhibits that you probably never imagined.  Nearly every exhibit is interactive in some way, and I don’t recall a single equation in the whole building.

For my inaugural blog entry, I’m going to write about one of my favorite exhibits in the museum.  It’s based on a seemingly simple game which conceals some surprising mathematical secrets.  I first heard about the game from Prof. Mel Hochster during the summer after I graduated from the University of Michigan in 2007.  He was teaching a summer course on the Fibonacci numbers to talented high school students (I was a course assistant) and he used the game to illustrate the notion of isomorphism, a term which mathematicians use to describe seemingly different phenomena which are secretly the same.

Anyway, here’s the game.  All you need is a friend, a piece of paper, and a pencil.  Write the numbers 1 through 9 at the top of the page, and take turns with your friend choosing a number, crossing out each number once it has been chosen.  The object of the game is to be the first person to select exactly three numbers which add up to 15. 

You pick 5, I pick 9.
You pick 4, I pick 6.
You pick 8, I pick 3.
You pick 2, and you win: 5 + 8 + 2 = 15! (Note that I didn’t win even though I picked 9 and 6 because I needed *exactly* three numbers which add up to 15.)

If you have a friend nearby, give the game a try. It’s a pretty challenging game, and it’s structure isn’t particularly obvious. Does either player have a winning strategy? Can either player force a draw? What is the best starting move? If you think about the game for long enough, you might eventually be able to provide answers to some or all of these questions, but it will probably take some effort.

The beautiful thing about the game is that it is completely equivalent (isomorphic, in fact!) to a much simpler game that almost everybody understands. To see what’s going on, we’ll use a so-called magic square:

Square 1

The property possessed by this table of numbers which makes it “magic” is that each row, column, and diagonal adds up to 15. Moreover, every triplet of numbers from 1 to 9 which add up to 15 is represented as some row, column or diagonal. Let’s go through the example game above one more time, only this time we’ll draw an “X” through each number that you picked and an “O” through each number that I picked. Here’s what it looks like at the end:

Square 2

As you can hopefully see, the game is nothing more than tic-tac-toe in disguise!

At the museum, they came up with a clever way to turn this game into an exhibit. The two players play the game on a computer screen, but one player is seated inside a concealed booth with a magic square. The poor player without the magic square has to labor through a lot of arithmetic, while the player in the booth just has to play tic-tac-toe.

I think this game (and the corresponding exhibit) is one of the best non-technical illustrations of what mathematics is all about.

  1. It demonstrates that two seemingly very different phenomena can be secretly “the same”.
  2. It shows how interesting abstract constructions (like the magic square) can be useful in unexpected ways.
  3. It shows how hard problems can become much simpler if you find the right way to look at them.

These are all extremely important themes in mathematics, and I hope to explore each of them further in future blog posts.