The box below forms a basis for a game between two players. The idea is that the two players take turns shading in one of the six rectangles (numbered 1 through 6) with one of two colors- say red or blue.
| | | |
| | 2 | 3 |
| | | |
|1| | |
| | 6 | 4 |
| | +---+
| | | 5 |
Either player can use either color on any turn. It is illegal to shade a rectangle with a color that has already been given to a neighboring rectangle. If you don't have a legal move at your turn, you lose the game.
Prove that for each opening move by the first player, the second player can always win.
Since 6 borders all the others, you don't want to go there first and once you've played anywhere else you can't choose 6.
By symmetry there are only 3 opening moves: 1, 2 or 3.
If player A chooses 1, Player B can choose 5, then A may choose 3 or 4, then B chooses 2 and there is no option for A.
If player A chooses 2, Player B can choose 4, then A must choose 5, then B must choose 1 and there is no option for A.
If player A chooses 3, Player B may choose 1 or 5, then A must choose the other, then either 4 or 2 is available to B and there is no option for A.
I could imagine larger boards for this game. In general it would appear the game favors player B. I wonder if there is a small board where the first player would win.
Edit: The above is incorrect. I didn't read the rules carefully and assumed each player was bound to a single color.
Edited on May 3, 2013, 11:08 am
Posted by Jer
on 2013-05-02 13:52:11