The Chocolate Bar
This is a level 2 question which should take approx. 4 minute in the interview
Two mathematicians are sharing a bar of chocolate whose lower left square is poisoned. They impose the following rule: when someone wants to take a square of chocolate, they must take the whole upper right corner with it. For example, on the figure below, the first took square 1, then the second took square 2.
Each takes their turn to take a square and the upper right corner that comes with it. The one who is forced to take the poisoned square (when it is the only square left) is declared the loser!
Show that there is a winning strategy for the player who begins.
(Of course, we don’t ask for the explicit winning strategy: this is an open problem)
Try to solve this problem yourself before moving on to the solution below. Use hint if needed
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Hint
What happens if the mathematician going second plays a winning strategy?
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Solution
Given that there is no possible equality, one of the two players (either the one who starts, or the other) has a winning strategy. Indeed, either the first player has a winning strategy, or the second player can always beat the first player regardless of the first player’s first move. In the second case, the second player has a winning strategy.
Assume, for a contradiction, that the second player has a winning strategy. If the first player begins by taking the square in the top right, the second player will find a move that, despite the first player’s best efforts, will lead to the second player’s victory. But, since the first move consisted only in taking the first square, the second move could have been played from the start by the first player. But then, in this situation, the second player is blocked; regardless of what the second player does, the first player will win (if they play optimally). This contradicts the fact that he has a winning strategy! It is thus the first player who possesses a winning strategy.
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😃 Your turn: Managed to crack this one?
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