The Gordon Game

The Gordon game is a game to be played by two people who know a bit about group theory (and by a bit, it’ll suffice to read my introduction to group theory). I found of the game in the book “Adventures in Group Theory: Rubik’s Cube, Merlin’s Magic and Other Mathematical Toys” by David Joyner.

According to the book, the rules are as follows. Take two copies of a set corresponding to the group and call one and the other . The rules are then as follows:

1. Player one chooses an element from the group and from . These can be any element he chooses, and once they are chosen they are removed from the sets and
2. The next player chooses an element and such that . Both elements are removed from their respective sets.
3. Repeat the previous step. The first player that cannot move loses.

I’ve written a program in C++ that will compile under GCC. The group used here is , that is, the integers modulo 7. You can download it here, although this link may change after I graduate. I wrote it hastily, so it’s not commented well and it probably has bugs. To compile, use the command

g++ gordon.cpp -o gordon

I can compile it fine on OpenSUSE 11.0. There’s no AI yet, so it simply switches between two human players. In the future, I may add support for user-defined groups and possibly some rudimentary AI.

Enjoy!

UPDATE: I updated the program a little bit so that you can make your own groups. As a result, you’ll now need to download the default file, z7z.grp, if you  want to be able to run the program.

To make your own groups, follow this recipe:

1. On the first line, list all of the group elements separated by a space and followed by an exclamation mark.
2. Now just list the multiplication table, with a space between columns and a new line between rows.

For example, the group looks like this:

0 1 2 3 4 5 6 !
0 1 2 3 4 5 6
1 2 3 4 5 6 0
2 3 4 5 6 0 1
3 4 5 6 0 1 2
4 5 6 0 1 2 3
5 6 0 1 2 3 4
6 0 1 2 3 4 5

To use your new group, pass it on as a command line argument, i.e. run

./gordon mygroup.grp

That’s it! Try it with a complicated group, for example a large order dihedral group.