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Two players $A$ and $B$ play the following game:

Start with the set $S$ of the first 25 natural numbers: $S=\{1,2,\ldots,25\}$.

Player $A$ first picks an even number $x_0$ and removes it from $S$: We have $S:=S-\{x_0\}$.

Then they take turns (starting with $B$) picking a number $x_n\in S$ which is either divisible by $x_{n-1}$ or divides $x_{n-1}$ and removing it from $S$.

The player who can not find a number in $S$ which is a multiple or is divisble by the previous number looses.

Is there a winning strategy?

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    It's a finite game, and draws are impossible, so of course there is a winning strategy (for one of the players). Perhaps what you are really asking is which player has a winning strategy, and what might a winning strategy be?2012-03-30
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    I believe there is a winning strategy for the second player. See my answer.2012-03-30

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