2013 AMC 12B Problems/Problem 18
Contents
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 or
coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove
or
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
coins and when the game starts with
coins?
Barbara will win with
coins and Jenna will win with
coins.
Jenna will win with
coins, and whoever goes first will win with
coins.
Barbara will win with
coins, and whoever goes second will win with
coins.
Jenna will win with
coins, and Barbara will win with
coins.
Whoever goes first will win with
coins, and whoever goes second will win with
coins.
Solution
We split into 2 cases: 2013 coins, and 2014 coins.
Notice that when there are
coins left, whoever moves first loses, as they must leave an amount of coins the other person can take. If Jenna goes first, she can take
coins. Then, whenever Barbara takes coins, Jenna will take the amount that makes the total coins taken in that round
. (For instance, if Barbara takes
coins, Jenna will take
). Eventually, since
it will be Barbara's move with
coins remaining, so she will lose. If Barbara goes first, on each round, Jenna will take the amount of coins that makes the total coins taken on that round
. Since
, it will be Barbara's move with
coins remaining, so she will have to take
coins, allowing Jenna to take the last coin. Therefore, Jenna will win with
coins.
If Jenna moves first, she will take
coin, leaving
coins, and she wins as shown above. If Barbara moves first, she can take
coins, leaving
. After every move by Jenna, Barbara will then take the number of coins that makes the total taken in that round
. Since
, it will be Jenna's turn with
coins left, so
Barbara will win. In this case, whoever moves first wins.
Based on this, the answer is
Video Solution
https://youtu.be/I6JgSPbL6gY ~Math Problem Solving Skills
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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