Let’s play a game. You name an integer from 1 to 10. Then we’ll take turns adding an integer from 1 to 10 to the number our opponent has just named, giving the resulting sum as our answer. Whoever reaches 100 first is the winner.
You go first. What number should you choose?
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Say a player is in position H if H is the number she’s named.
Any player in position 90-99 is lost, because her opponent can advance immediately to 100.
That means that a player in position 89 will win, since her opponent will be forced to enter this losing territory.
But then position 78 is also winning, since the opponent must open a road to 89.
And so on backward: 67, 56, 45, 34, 23, 12, and 1 are all winning positions.
With correct play, then, the first player always wins: She chooses 1 and then aims for each of these milestones.
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