Reach 500!

Logic Level 2

Dan and Sam play a game in which the first to start says the number 1, the next says 2, and the one who's next must say an integer number between the number previously said and its double (but not including).

For example, Dan begins saying 1, then Sam says 2, and then Dan can say whichever number he wants between 2 and 4; as the only integer between 2 and 4 is 3, he must say \(3\). Then, Sam can choose any number between 3 and 6; that is, he can say either 4 or 5.

The game finishes when someone reaches 500 (who is the winner). If Dan begins, what number must say the one who has a winning strategy, in his \({ 5 }^\text{th}\) turn?

Assume both players play optimally.

This is the fifth problem of the set Winning Strategies.

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