The next two exercises deal with the game of Euclid. Two players begin with a pair of posi...

The next two exercises deal with the game of Euclid. Two players begin with a pair of positive integers and take turns making moves of the following type. A player can move from the pair of positive integers {x, y} with x ≥ y, to any of the pairs {x − ty, y}, where t is a positive integer and x − ty ≥ 0. A winning move consists of moving to a pair with one element equal to 0.

Show that in a game beginning with the pair {a, b}, the first player may play a winning strategy if a = b or if ; otherwise, the second player may play a winning strategy. (Hint: First show that if , then there is a unique move from {x, y} that goes to a pair {z, y} with