Static Nim Game?

Question

Pile of five stones: “Two players, x and y , take turns removing stones from a pile.

The pile has five stones in the beginning, and each player can remove one or two stones in each turn. Whoever gets the last stone wins. x goes first.”

How can i give the decision tree, the strategies for both players and the decision matrix of this game.

Answer 1

Whoever starts a turn when there are 3 stones left will lose (if they both play logically).  X should pick up 2 stones.  If X picks up 1, Y should pick up 1, to leave 3.

If there are more than 5 to start with, you should pick up either 1 or 2 so that there are a multiple of 3 stones left, as soon as you can. (If there are a multiple of 3 already, you can't do it.)  From then on, if the other player gets 1, you pick up 2, and vice versa.