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Alberto and Barbara play a game

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November 9, 2010
inductioncombinatorics unsolvedcombinatorics

Problem Statement

Let NN be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing 11, and thereafter if a player has written nn on a certain move, his adversary is allowed to write n+1n+1 or 2n2n as long as he/she does not obtain a number greater than NN. The player who writes NN wins. (a)(a) Determine which player has a winning strategy for N=2005N=2005. (b)(b) Determine which player has a winning strategy for N=2004N=2004. (c)(c) Find for how many integers N≤2005N\le 2005 Barbara has a winning strategy.
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