Question

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1. Consider the following game: suppose there are three piles of stones starting with 3, 5, and 7 in each pile. Two players take turn playing this game. During their turn, a player can choose one of the three piles that are currently nonempty and remove any positive number of stonescurrently presented in that pile. Whoever takes the last stone loses the game.

(a) Describe how to model this game using a graph, in such a way that paths in your graph represent the different states that you can have in this game. Specify whether your graph is directed or undirected. Carefully describe what the vertices of your graph represent, and when two vertices are connected with an edge.

b) Suppose Player 1 (who makes the first move in this game) just won. What can you say about the path from (3, 5, 7) to (0, 0, 0) in this graph?

c) Suppose the game currently has 1 stone in the first pile, 2 stones in the second, and 2 stones in the third, and it is Player 1’s turn to make their move. Describe a winning strategy for this player.

Answer 1

a) We can model this game using a graph by plotting it as the players take their moves, and it will be directed towards (0,0,0) where the game ends.

The only time the vertices will be connected with the edge is at the time when a pile becomes empty.

b) Player 1 won, means the last move was made by player 2. Which means the total number of moves are even, telling us that the no. of vertices in our graph in even.

c) If player 1 takes 1 stone from any pile containing 2 stones,

Now the piles contain 1,1 and 2 stones.  

Case 1: Player 2 picks up 1 stone from a pile with 1 stone.

Case 2: Player 2 picks up 1 stone from the pile with 2 stones.

Case 3: Player 2 picks up 2 stones from the pile with 2 stones.

In case 1, player one can just take 2 from the 2 stone pile and leave player 2 with the last stone to pick.

In case 3, player one can just pick any one stone, leaving only one for Player 2

But in case 2, piles are at 1,1,1 which means player 1 will lose.

So Player 1 has to pick a stone from the second or third pile, and hope for Case 1 or 3 mentioned above.