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Given a $9 \times 25$ chessboard, a rook is placed at the lower left corner. Players A and B take turns moving the rook. A plays first and each turn consists of moving the rook horizontally to the right or vertically above. The last person to make a move wins the game. At the completion of the game, the rook will be at the top right corner. For example, the figure below shows a $3 \times 4$ chessboard and the sequence of moves that leads to a win for player A. Does player A have a winning strategy in the given $9 \times 25$ chessboard? If so, what is the strategy? If not, what is player B's winning strategy? 3x4 Game

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This is Nim in disguise. You have piles of $8$ and $24$. $A$ can win by symmetry.

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  • $\begingroup$ How it's possible explain me in detail $\endgroup$ – Bug_Hunter Jan 14 '18 at 16:47
  • $\begingroup$ Did you look up Nim? You have piles of stones and can take as many stones from one pile as you like. The horizontal and vertical distances are your two piles. You start 8 squares down and 24 squares left $\endgroup$ – Ross Millikan Jan 14 '18 at 16:49
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In an $n \times m$ game $(n\gt m)$ (the rook begins at $(0,0)$), $A$ has a winning strategy:
On your first move, you go to $(n-m,0)$. The winning strategy is then to end on the diagonal at the end of your turn.
If you play a $n \times n$ game, $B$ has the exact same winning strategy.

In general, the first one the come on a $(n-k,m-k)$ tile wins.

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