Reach 200

Logic Level 3

Dan and Sam play a game. Dan starts and says the integer \(1\), then Sam says \(2\). For each subsequent turn, the one who's next must say an integer that is strictly between the integer previously said and its double.

For example, Dan begins saying \(1\), then Sam says \(2\), and then Dan can say whichever integer 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 integer between \(3\) and \(6\); that is, he can say either \(4\) or \(5\).

The person who first says \(200\) is the winner. For the person who has a winning strategy, what is the minimum number of integers that he says?

Submit \(0\) if you think that nobody has a winning strategy.

Assumptions: Both players want to win the game.

This is the fourth problem of the set Winning Strategies.

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