AbigailII: Of course :-) For game n, the first n moves are moves 1 thru n of the infinite game. You then pick the shortest sequence to finish the game. Call the (n+1)th move X. X may or may not be the (n+1)th move in the infinite game. However, there will be an infinite number of values of n for which X is also the (n+1)th move in the infinite game. Therefore, the shortest sequence that finishes game n will also be the shortest sequence that finishes game (n+1).