Section 2.8.1 Alice and Bob play a game where they alternate moves, and Alice st

Question

Section 2.8.1

Alice and Bob play a game where they alternate moves, and Alice starts. There is a circle of 12 coins, and on each move a player can remove either one coin or two coins that were originally adjacent on the circle. For example, on Alice's first move she might remove the coin at position 3. Then Bob might remove the consecutive pair of coins at positions 8 and 9; but Bob may not remove the two coins at positions 2 and 4, because they were not originally adjacent. The player who removes the final coin wins.

Q: 2.74 The n coins in a circle game is the same as the 12 coins game from Section 2.8.1, except that the number of coins initially in the circle is any natural number n. For what values of n does Alice have a winning strategy? For what values does Bob have a winning strategy? Prove your conjectures.

Explanation / Answer

Alice is starting the game, for n be even or odd number, suppose take n be an even number If Alice make a first move then there are n-1 coints left, that count will be an odd and forms an arc like form and every odd count had middle one number bob can choose the middle coin in the arc, by following symmetry based on alice move bob can never loose the game. If Alice takes two coints also for n be an even number, bob will have even number of choice so that he can move middle two coins in the arc and same here also by following the symmetry bob can never lose the game. Alice will win when n=1 and n=2 because she can pick one or two coins at a time.

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