In each move, one player cuts the string between two pearls and the other player chooses one of the resulting parts of the string while the other part is discarded.
In the first move, Alice cuts the string, thereafter, the players take turns.
A player loses if he or she obtains a string with a single pearl such that no more cut is possible.
Who of the two players does have a winning strategy?
Solution
We claim that the winning situations are exactly the strings of an even number of pearls. We prove this claim by induction.
A string with one pearl is a losing situation by definition.
A string with an even number of pearls can easily be cut into two odd parts. These parts are a losing situation for the other player by induction, so that is a winning situation.
For an odd number of pearls, each cut produces an even part. The other player can thus choose this even part, which is a winning situation by induction. Therefore, the odd number is a losing position.
We conclude that Bob has a winning strategy.
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