Thanks for all your help.

By the way, the original problem that started me thinking about this was
one of the puzzles given at the MoMath masters tournament last week:

Imagine a two-player "game", where player A takes a number (integer),
doubles it, and passes it to player B, who adds 50 and passes it back to
player A. The first player to name a number greater than 1000 loses. If A
starts the game by passing an initial number to B, what's the lowest number
A can give that guarantees a win?

Regards,
Johann
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