Problem:
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara's turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with 2013 coins and when the game starts with 2014 coins?
Answer Choices:
A. Barbara will win with 2013 coins, and Jenna will win with 2014 coins.
B. Jenna will win with 2013 coins, and whoever goes first will win with 2014 coins.
C. Barbara will win with 2013 coins, and whoever goes second will win with 2014 coins.
D. Jenna will win with 2013 coins, and Barbara will win with 2014 coins.
E. Whoever goes first will win with 2013 coins, and whoever goes second will win with 2014 coins.
Solution:
If the game starts with 2013 coins and Jenna starts, then she picks 3 coins, and then no matter how many Barbara chooses, Jenna responds by keeping the number of remaining coins congruent to . That is, she picks 3 if Barbara picks 2, and she picks 1 if Barbara picks 4. This ensures that on her last turn Jenna will leave 0 coins and thus she will win. Similarly, if Barbara starts, then Jenna can reply as before so that the number of remaining coins is congruent to . On her last turn Barbara will have 3 coins available. She is forced to remove 2 and thus Jenna will win by taking the last coin.
If the game starts with 2014 coins and Jenna starts, then she picks 1 coin and reduces the game to the previous case of 2013 coins where she wins. If Barbara starts, she selects 4 coins. Then no matter what Jenna chooses, Barbara responds by keeping the number of remaining coins congruent to . This ensures that on her last turn Barbara will leave 0 coins and win the game. Thus whoever goes first will win the game with 2014 coins.
The problems on this page are the property of the MAA's American Mathematics Competitions