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Two players, one point, one clock, no time limit, who wins?
Imagine two players, Bob and Joe.
Bob has an integer $b$;
and Joe has an integer $j$;
Bob keeps on adding $b$ to $j$, until he notices Joe has won.
If Bob continues adding $b$ to $j$, Bob wins, and if Joe continues adding $b$ to $j$, then Joe wins.
If neither player stops playing after a certain number of turns, then Joe wins.
Do you know how long the game should last for, in order for Joe to win?
A:
Assuming that $b=1$ and that the board is always finite, the game has to end if $j=0$, otherwise either player can win. This can be seen as follows:
If $j=0$ then $j+b=b$, so Bob always wins
If $j>0$ then $j=j+b-b<j+b$ so Joe wins.
This can be generalized to any $b\ge 2$ for an arbitrary finite board where Joe must stop playing in finite time, or else Bob wins. This is also true for a more general version of the game, where instead of Bob playing one number, Bob plays an integer $k\ge 2$.
If you change the game so that it is only played once, and
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