Pass 300

Logic Level 5

Let \( k \) be a positive integer. Dan and Sam play a game in which the first to start says the number \(k\) and the one who's next must say a multiple of the previous number, that is between the previous number and its square. They cannot repeat a number even if the number was said by the other. Also, the said number cannot be greater than 300.

For example, Dan begins saying \(3=k\), then Sam can reply 6 or 9, but not 3, because Dan said it before.

The winner is the one who cannot say a number in his turn. If Dan begins, and both players play optimally, for how many numbers \(k\le 200\) does Dan win?

This is the nineteenth problem of the set Winning Strategies.

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