A clown approaches you, holding 99 red balloons. He proposes a game where the two of you will take turns popping some nonzero number of balloons. The only restrictions are:
- On the first turn, the current player can't pop all the balloons.
- On later turns, a player can't pop more balloons than their opponent just popped.
Whoever pops the last balloon wins.
The clown gives you the choice of going first or second. Which do you choose, and how do you win?
Answer
A strategy is:
Go first.
Pop one balloon.
Because
There must be a non-zero number of balloons popped and neither player can pop more than one balloon per turn $\implies$ exactly one balloon is popped each turn, and the odd-numbered player (first player) wins.
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