A game is being played with two players. There are 21 balloons and the player who pops the last balloon wins. One or two can be popped in a turn. How can a player win every time

Question

Dempsey

Mathematics

28 April, 17:51

A game is being played with two players. There are 21 balloons and the player who pops the last balloon wins. One or two can be popped in a turn. How can a player win every time

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Answers (1)

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Francisco Santana · Answer 1

be the second player, and always leave a multiple of 3 balloons

Step-by-step explanation:

In order to win, a player must force the other player to leave one or two balloons. To do that, the winning player must leave one more balloon than the maximum number that can be popped. That is, the winner will be the player who leaves 3 balloons,

Working backward, we find that the winner must leave a multiple of 3 after each turn. Since the starting number is a multiple of 3, the first player must lose if the second player plays optimally.

The winning strategy is ...

be the second player always leave a multiple of 3 balloons.