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Grid nim: 2xN (Posted on 2019-01-16) Difficulty: 3 of 5
Two players start with a 2xN grid of squares. Each player, in turn, may take either a single square or a 2x2 block of squares. The player who takes the last square loses.

For what values of N does player 2 win?

No Solution Yet Submitted by Jer    
Rating: 4.5000 (2 votes)

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Question Spoiling moves | Comment 1 of 4
In cases 1*2 and 2*2, Player 1 always wins by playing 'small' i.e. taking 1*1 squares.

On one view, 2*3 is a win for Player 2, because if Player 1 plays 'big' i.e. takes a 2*2 block then Player 2 takes one of the 2 remaining, leaving Player 1 with the last, but if Player 1 plays 'small' then Player 2 plays 'big', again leaving Player 1 with the last.

If however Player 1 is able to 'spoil' by removing a single square leaving no complete 2*2 block, then of course Player 1 will win. Is this supposed to be possible? If so, it seems doubtful that Player 2 can ever win. Or is Player 2 allowed to remove 4 squares to make a 2*2 block, whether or not they were in that configuration on the grid?

  Posted by broll on 2019-01-17 04:20:43
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