Not to Reach 10000

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 strictly between, (not including the endpoints), the number previously said and its square. Also, the said number can't be greater than the goal number, that is, 10000.

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 9; that is, he can say either 4, 5, 6, 7 or 8.

The game finishes when someone reaches 10000 (who is the loser). If Dan begins, who will win? This means, who has a winning strategy?

This is the eighth problem of the set Winning Strategies.

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