Two players take turns taking sticks from a pile of 16 sticks. Each player can take...

Two players take turns taking sticks from a pile of 16 sticks. Each player can take at most 3 sticks and at least 1 stick at each turn. Whoever takes the final stick wins the game. Describe in words the optimal strategy for each player. Is there a first-mover advantage in this game? Is there a second-mover advantage?

Expert Solution

There is a second-mover advantage in this game. If played according to the best strategy, the second mover will always win this game. Let's see why.

Le's do some backward induction. the player picking the 16th sick wins. Now, a player can guarantee to pick the 16th stick if it picks the 12th stick ( because if A player picks the 12th stick, the B player can pick a max of 3 sticks and min of 1 stick, thus A player can pick the 16th stick in the next round because the max sticks remaining are 3 and min sticks remaining is 1). Similarly, in order to ensure that a player picks the 12t stick, he must pick the 8th stick and so on till the player has to pick the 4th stick. Now the first mover can not reach the 4th stick in the first chance, but the second mover can, and after that following the strategy he can reach the 8th, 12th, and 16th stick easily.

So the optimal strategy for player 2 is 4 - number of sticks taken by player 1.

Player 1 will always lose. The best it can do is hope for player 2 to miss the best strategy once and then it plays similarly to palyer 1

Rahul Sunny answered 8 months ago

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