This month’s puzzle proved pretty tricky, and we received many submissions which were very close but not quite right. The smallest 3 numbers for which Bob will win the game are 11, 22, and 32. In fact, for larger numbers, Bob will win if N is congruent to 11, 22, or 0 mod 32.

*Alice can obviously win for N=1 up to N=9, and she can also win when N=10 if she says 5 (forcing Bob to pick something other than 5).

*When N=11, Bob wins: If Alice picks some k>1, Bob will be able to pick 11-k. If Alice picks 1, Bob will be able to pick 5.

*For N = 12 up to N = 20, Alice can win by picking the number which gives Bob the tally of N-11. For N = 21, Alice can also win if she picks 5, since Bob must pick something other than 5. If he picks anything other than 8, Alice can win easily. If he picks 8, Alice can counter with 4, Then Bob with 2, then Alice with 1, and thus Alice wins anyway.

*When N = 22, Bob wins, for similar reasons as when N = 11.

*When N = 32, Bob wins as well. If Alice says any k other than 5, Bob can say 10-k. If Alice says 5, Bob can say 8, leaving Alice with a remaining sum of 19 but unable to say 8, which again forces a Bob win.

Congratulations to everyone who solved this month’s puzzle!