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Alice and Bob play the following (base 10) number game. A target N is fixed, N being a positive integer. Alice then writes the number 1 on the blackboard. Bob responds with the number 2. Thereafter, at each of their alternating turns, a player responds by adding to the last number written by the other player any of the nonzero digits of that number. The first player to reach or surpass the target N wins.

It has been conjectured that for N > 36, Bob can always win. Can anyone prove this? The question has been dealt with at:

https://math.stackexchange.com/questions/2985890/a-number-game-for-two/2992723#2992723

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