Question 695384: There are two stacks of cards that each contain "r" cards. Two players play the following game. Each player in turn chooses one stack and then removes any number of cards, but at least one, from the chosen stack. The player who removes the last card wins the game. Prove that the second player (the player that does not go first) can always win. Don't prove by example cover all cases.
Found 2 solutions by Edwin McCravy, solver91311:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website! There are two stacks of cards that each contain "r" cards. Two players play the following game. Each player in turn chooses one stack and then removes any number of cards, but at least one, from the chosen stack. The player who removes the last card wins the game. Prove that the second player (the player that does not go first) can always win. Don't prove by example cover all cases.
Strategy: The second player simply always causes the two
stacks to have the same number of cards.
Explanation:
The object is to force your opponent to take all the cards
in one of the stacks. For then you can pick up last by
taking the other stack. Let's assume the first player is
not stupid enough to lose early by taking all the cards in
one of the piles until forced to do so.
1. First player has no choice but to leave the two stacks with a DIFFERENT
number of cards.
2. Second player then takes the same number of cards the first player took
off of one pile, off the other pile, leaving the two stacks with the SAME
number of cards.
3. Then the first player must again leave the two stacks with a DIFFERENT number of cards.
4. Second player then takes the same number of cards the first player took
off of one pile, off the other pile, leaving the two stacks with the SAME
number of cards.
5. Then the first player must again leave the two stacks with a DIFFERENT number of cards.
etc. etc.
This continues until the second player leaves the two stacks with just 1
card in each pile. Then the first player must take 1 card, and the second
player picks up last and wins.
Edwin
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
Player 2 wins every time by taking the same number of cards on each draw from the opposite pile from that chosen by player 1. Regardless of the number of cards taken on any turn by player 1, s/he must ultimately take the last card in one of the stacks. Once that happens, player 2 takes the remainder of the cards in the other stack and wins.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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