Cut-Chocolate
Heidi and Victor have a chocolate bar shown below:
To decide who gets the chocolate, they play a turn-based game, without changing the rectangular configuration of the chocolate bar:
- In Heidi's turn, H eidi breaks any of the pieces along a h orizontal line.
- In Victor's turn, V ictor breaks any of the pieces along a v ertical line.
- Whoever runs out of moves first loses.
Assuming that both Heidi and Victor play optimally, who wins or loses the game?
Inspired by the book Winning Ways for your Mathematical Plays
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If Heidi starts, his one horizontal break leaves no horizontal breaks to make. Victor breaks one of the pieces vertically, it does not matter which one and where, and then Heidi can't make a move and loses.
If Victor starts, he will make one small piece (two squares stacked on top of each other) and one larger one (four squares). Heidi needs to break the smaller piece in two. Victor then has no choice but to break the larger piece into two small ones, each two squares stacked on top of each other. But that uses up all the vertical breaks to be had. Heidi has a move, either piece will do, but Victor does not.
The person starting loses in all cases, if the game is played well by both players.