Saturday, July 30, 2005

A fascinating correction on the 2x2 board

Infinitum no itte?

David Fifield contacted me regarding my original comment on the 2x2 board that "the first one who plays loses". His explanation is rather fascinating and I've reproduced it here in its entirety. Blogger has a bad habit of reformatting the text so you'll have to examine the diagrams carefully as you read.


The outcome of the 2 x 2 game depends on the ruleset used, but perfect play always leads to either a repeating position, a draw, or a win for Black.

The only way the game is guaranteed to end is with superko, so this
example uses that.

0 1 2 3 4
----- ----- ----- ----- -----
| . . | | . X | | . X | | . X | | O . |
| . . | | . . | | O . | | O X | | O . |
----- ----- ----- ----- -----

1. All four opening move are equivalent when rotation is considered.
2. This is White's only move. Anywhere else leads to life for Black with two eyes.
3. Here Black puts himself in atari...
4. And White captures.

But the game isn't over. Black now has four options: to resign, to pass,or to play on one of the two remaining points. Resignation an passing lead to an immediate loss. Playing at the same point as the first move also loses for Black:

5 6
----- -----
| O X | | O . |
| O . | | O O |
----- -----

Now superko prevents Black from capturing, because that would recreate position 1, so White wins.

Consider what happens, though, if Black plays at the other point:

5 6 7 8 9
----- ----- ----- ----- -----
| O . | | O O | | . . | | O . | | O X |
| O X | | O . | | . X | | . X | | . X |
----- ----- ----- ----- -----

7. Snapback!
8. White must play here for the same reason she had to play 2.

After position 9, superko prevents White from capturing and recreating position 4. Now, with territory scoring, both players have 0 territory and 3 captures. With area scoring, Black wins by 1. You can see that without superko, both players can just keep capturing and recapturing forever.

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