The Math of Games

On Mon, November 28, 2011 8:51 am, [a friend] wrote [to the MathFuture mailing list]:

Hello, I have recently started an after school program for teenagers. The main topic is algebra.  We are now working with polynomials. I am looking for math games that let students practice algebra while interacting in a game-like way with math.

Can you give me any suggestions about it ?

I replied:

I recommend looking into Winning Ways for Your Mathematical Plays, which is based on John Horton Conway’s On Numbers and Games. There is software for manipulating the resulting numbers and games, which have a polynomial-like structure involving standard numbers and infinitesimals. Roughly speaking, a number in this system is a game with a constant value, known in the jargon as a cold game. Hot games are those where the score changes as players make their moves.

In Conway games the first player without a move loses. The simplest game is 0 ({|}, the ordered pair of two empty sets), the game in which neither player has a move, and so the player to move loses. In chess terminology such a situation is called zugzwang (tsooktsvank).

The game where player 1 loses instantly if he has to move but player 2 can move to 0, and thus wins instantly if it is her turn, has the value -1 ({|0}), and in the reverse case ({0|}) we get 1. The game in which the player to move wins ({0|0}) is called *. It is not less than 0, equal to 0, nor greater than 0. *+*=0, because the first player wins one of the * games, and the second player wins the other and thus the combined game. Since the first player to move loses, this is by definition equivalent to 0. There is also an addition rule for games that gives the same result. And so on through four volumes, and several other books applying these techniques to dots and boxes, go, and other games.

If past performance is any guide, it would take a century to get these ideas into curricula in the usual manner. I am therefore trying to short-circuit or bypass the usual manner.

Thanks !

Always a pleasure.

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About mokurai

Generalist; End poverty at a profit for all
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