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.
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  2  3 
   
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1  
  6  4 
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   5 
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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.
Call the players A and B for convenience.
By color symmetry, although A can use either color, I will call the first color used Red.
By board symmetry, a first move of 1 is the same as a first move of 5 and 2 is the same as 4.
So we need to consider four of the 12 possible opening moves:
A plays red on 1,2,3or6
Case A plays red 1:
B plays 5 blue. 6 becomes unavailable.
If A plays 2 blue or 4 red, B will be left with the final move of playing the opposite color in either of the remaining spaces.
If A plays 3 in blue or red, B will be left with the final move of playing the opposite color in either of the remaining spaces.
Case A plays 2 red.
B plays 4 blue. 6 becomes unavailable.
The are only two possible moves left: 1 blue or 5 red
when A chooses one B gets the other.
Case A plays 3 red.
B plays 6 blue.
There are only two possible moves left: 1 red or 5 red.
When A chooses one B gets the other.
Case A plays 6 red.
B plays 3 blue. (This becomes the same as above, with opposite colors.)
There are only two possible moves left: 1 blue or 5 blue.
When A chooses one B gets the other.

Posted by Jer
on 20130503 11:21:21 