Wild About Math! Making Math fun and accessible

A fun arithmetic game that sparks exploration

Here's a game that's easy and leads to a nice exploration of number theory for those so inclined. Two people play. All you need is a sheet of paper and a pencil or pen. Here's how to play:

1. Each person thinks of a number between 1 and 50 without telling the other person what the number is. Then, each person writes their number on the sheet of paper.
2. Decide who is going to go first, by tossing a coin or in some other mutually agreeable way.
3. Players take turns writing down the positive difference between any two numbers on the sheet of paper.
4. Numbers cannot appear more than once on the paper.
5. The player who cannot write down a unique positive difference loses.

Here's an example of how a game might go between Sol and his friend Michele.

1. Sol thinks of the number 5. Michele thinks 3.
2. They write 5 and 3 on the paper.
3. Sol goes first.
4. 5 minus 3 is 2 so Sol adds 2 to the paper.
5. The paper now has these numbers: 5 3 2
6. Michele notices that 5 minus 2 is 3 but 3 is already on the paper.
7. Michele also notices that 3 minus 2 is 1 so she writes 1 on the paper.
8. The paper now has these numbers: 5 3 2 1
9. Sol notices that 5 minus 1 is 4. He writes 4 on the paper.
10. The paper now has these numbers: 5 3 2 1 4
11. Sol wins as no more unique differences can be calculated.

Here's another sample game:

1. Sol thinks 8. Michele thinks 6.
2. The paper has: 8 6
3. Michele goes first.
4. Michele notices that 8-6=2. The paper now has: 8 6 2
5. Sol notices that 6-2=4. The paper now has 8 6 2 4.
6. The game is over and Sol wins as no more unique differences can be calculated.

Here are some interesting exploration questions:

1. Once both numbers are written down is there a way to determine who will win?
2. Once both numbers are written down does strategy matter, other than who goes first?
3. For starting numbers of 5 and 3 all numbers between 1 and 5 got written down but when 6 and 8 were the starting numbers only 2, 4, 6 and 8 were possible differences. What determines whether all numbers get used and if not which ones are used and which aren't?

This game is related to Euclid's algorithm and to the greatest common divisor of two integers. At Cut the Knot there's a Java version of this game, Euclid's Game, that you can play alone against the computer. In the computer game the computer picks the two starting number but you can practice determining who should go first.