55rd Ukrainian National Mathematical Olympiad - Third Round

Question

Winnie-the-Pooh and Piglet play the following game. There is a 15-inch-long stick. By his first move, Piglet breaks it into two pieces, then the players in turn break one of the existing pieces into two. The rules are that the resulting pieces must have integer length (in inches) and can't be 1-inch-long. The player who can't make a move loses. Who has a winning strategy?

Accepted Answer

15 = 3+3+3+3+3 = 3+3+3+2+2+2 = 3+2+2+2+2+2+2. This implies that in order to win Piglet needs to ensure the existence of two 3-inch and one 2-inch pieces (this will make the first and the third outcomes impossible). So, by his first move he must break the stick into 5 and 10 inches. The first piece will be eventually divided into 2 and 3 inches. If Winnie breaks the 10-inch stick by his next move, Piglet must break 3 inches off the longer part, otherwise he should just break it into 3 inches and 7 inches. In this case, regardless of Winnie's moves, in the end there will be at least two 3-inch pieces and one 2-inch piece, which is enough for Piglet to win.