Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either one, two, or three matchsticks. The player to remove the last matchstick wins the game. (a) Suppose there are only 5 matchsticks left, and it is Fred’s turn. What move should Fred make to guarantee himself victory? Explain your reasoning.
(b) Suppose there are 10 matchsticks left, and it is Fred’s turn. What move should Fred make to guarantee himself victory?
(c) Now start from the beginning of the game. If both players play optimally, who will win?
(d) What are the optimal strategies (complete plans of action) for each player?
Hope your writing looks readable and please draw the tree if possible.