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You and your friend are playing the following (fun!) game. There are two piles of apples. The first player removes some positive integer number of apples from one of the piles. Then, the second player also removes some number of apples from one of the piles. Each player can remove any number of apples from one of the two piles, but cannot take apples from both piles simultaneously. The players alternate until one of them removes the very last apple (from either pile). That player, who removes the very last apple, wins. You deviously encourage your friend to go first. Then, you use a simple copycat strategy: whenever your friend removes j apples from one pile, you always remove the same number japples from the second pile. Prove the following statement using strong induction: Proposition: If at the beginning of the game the two piles contain the same number of apples, then you (the second player) always win using your copycat strategy.

You and your friend are playing the following (fun!) game. There are two piles of apples. The first player removes some positive integer number of apples from one of the piles. Then, the second player also removes some number of apples from one of the piles. Each player can remove any number of apples from one of the two piles, but cannot take apples from both piles simultaneously. The players alternate until one of them removes the very last apple (from either pile). That player, who removes the very last apple, wins. You deviously encourage your friend to go first. Then, you use a simple copycat strategy: whenever your friend removes j apples from one pile, you always remove the same number japples from the second pile. Prove the following statement using strong induction: Proposition: If at the beginning of the game the two piles contain the same number of apples, then you (the second player) always win using your copycat strategy.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me solve this problem about induction. 

Problem 1 You and your friend are playing the following (fun!) game. There are two piles of apples. The first player removes some positive integer number of apples from one of the piles. Then, the second player also removes some number of apples from one of the piles. Each player can remove any number of apples from one of the two piles, but cannot take apples from both piles simultaneously. The players alternate until one of them removes the very last apple (from either pile). That player, who removes the very last apple, wins. You deviously encourage your friend to go first. Then, you use a simple copycat strategy: whenever your friend removes j apples from one pile, you always remove the same number j apples from the second pile. Prove the following statement using strong induction: Proposition: If at the beginning of the game the two piles contain the same number of apples, then you (the second player) always win using your copycat strategy.
Transcribed Image Text:Problem 1 You and your friend are playing the following (fun!) game. There are two piles of apples. The first player removes some positive integer number of apples from one of the piles. Then, the second player also removes some number of apples from one of the piles. Each player can remove any number of apples from one of the two piles, but cannot take apples from both piles simultaneously. The players alternate until one of them removes the very last apple (from either pile). That player, who removes the very last apple, wins. You deviously encourage your friend to go first. Then, you use a simple copycat strategy: whenever your friend removes j apples from one pile, you always remove the same number j apples from the second pile. Prove the following statement using strong induction: Proposition: If at the beginning of the game the two piles contain the same number of apples, then you (the second player) always win using your copycat strategy.
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