Two people take turns removing stones from a pile of n stones. Each person may, on each of her turns, remove either one or two stones. The person who takes the last stone is the winner; she gets $1 from her opponent. Find the subgame perfect equilibria of the games that model this situation for n = 1 and n = 2. Find the winner in each subgame perfect equilibrium for n = 3, using the fact that the subgame following player 1's removal of one stone is the game for n = 2 in which player 2 is the first mover, and the subgame following player 1's removal of two stones is the game for n = 1 in which player 2 is the first mover. Use the same technique to find the winner in each subgame perfect equilibrium for n = 4, and, if you can, for an arbitrary value of n.