# The Chocolate Game

Please comment your solutions, questions and remarks..

I heard this from Lauri Hella, but he doesn’t remember from whom he heard this.

The game is played between two players as follows. There is an $$n\times m$$ bar of chocolate on the table. See the picture below. First Player 1 chooses a chocolate piece and then removes (eats!) all the pieces that lie up-right from that piece. For instance if he picks the piece (4,2), then he removes all the pieces $$(x,y)$$ such that $$x\geqslant 4$$ and $$y\geqslant 2$$. This move is illustrated in the picture. Then Player 2 does the same: she chooses a piece, say (5,1), and removes all pieces up-right from it (in this case only one, since (5,2) and (5,3) are already removed by Player 1). This move is not illustrated. At each move a player has to remove some chocolate in this way. The player who takes the last piece $$(0,0)$$, loses.

Show that Player 1 (the beginner) has a winning strategy. (Assume nm>1)

Level 3/5

The picture is taken from here, is made by Kifer and is edited by me (using gimp..). It is licensed under a Creative Commons license

Hint after Read More

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WARNING, a hint below.
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Use similar reasoning as in the proof of Theorem 2 in the post about tic-tac-toe

## About Vadim Kulikov

For details see this
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### 2 Responses to The Chocolate Game

1. This game is called Chomp:
http://en.wikipedia.org/wiki/Chomp
(Added by Vadim: Warning, there is a solution behind the link above).

2. Thank you. I didn’t know it.