A step by step analysis is given for the following questions,

1. Find an upper bound for the number of possible states in the game of chess, assuming that draw-by-repetition is enforced if the same position is repeated three times.

2. Find an upper bound for the number of possible moves in a single turn in the game of chess.

3. Use Question 2 above to find an upper bound for the number of possible sequences of moves in 10 consecutive turns in the game of chess.

4. Chess experts sometimes think 10 turns ahead, but do not consider every possible sequence of moves. Given a computer capable of processing one billion moves per second, find the maximum number of options it could consider at each move if it must think 10 turns ahead within a 3-minute time period. [Note: what we have called a "turn" is actually called a "half-move".]