Andrew and Betty play a game in which they lay out a row of three coins, heads up. They take turns, begining with Andrew, turning over one of the coins at a time. They must not produce a pattern of heads and tails which has already occurred earlier in the game. The first person who cannot make a move is the loser.

1. If they each play as well as possible, who is the winner?

2. If the game were played with four coins instead of three, who would be the winner?

3. If the game is played with three coins but the player who cannot make a move is declared the winner, who wins now?

Think of the possible states of the game as vertices of a cube, with coordinates like (H,H,H), (H,H,T), (H,T,H), and so on up to (T,T,T). You start at the (H,H,H) corner, and each move goes along a side to another vertex.

For the second question, imagination gets a workout, for you should think about movements along the sides of an hypercube.

Posted by Old Original Oskar! on 2006-09-13 13:15:19