Consider the following two-player game. There are stones placed on a table. Players alternate turns. On...

 

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Consider the following two-player game. There are stones placed on a table. Players alternate turns. On a player's turn, they must remove k stones for some 0 <k<n such that k divides n, where n is the number of remaining stones on the table. If there are no available moves for the player, that player loses. Let us say the players are Alice and Bob and that Alice goes first. An example game that starts with 9 stones is: n 9 6 4 3 2 1 Player Alice Bob Alice Bob Alice Move 3 2 1 1 1 Bob 0 In this example game, Bob loses. However, neither player played optimally! Assuming both players play optimally, provide a rule that, given n, determines whether Alice wins. Prove the correctness of this rule via mathematical induction. Hint: a move is always available except when n = 1. Consider the following two-player game. There are stones placed on a table. Players alternate turns. On a player's turn, they must remove k stones for some 0 <k<n such that k divides n, where n is the number of remaining stones on the table. If there are no available moves for the player, that player loses. Let us say the players are Alice and Bob and that Alice goes first. An example game that starts with 9 stones is: n 9 6 4 3 2 1 Player Alice Bob Alice Bob Alice Move 3 2 1 1 1 Bob 0 In this example game, Bob loses. However, neither player played optimally! Assuming both players play optimally, provide a rule that, given n, determines whether Alice wins. Prove the correctness of this rule via mathematical induction. Hint: a move is always available except when n = 1.

Answer rating: 100% (QA)

The Problem The picture shows stones on a table and its a twoplayer game Players remove stones in accordance with a set rule taking turns The player who is unable to complete a move wins the game Guid... View the full answer