2013 AMC 12B Problems/Problem 18

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?

$\textbf{(A)}$ Barbara will win with $2013$ coins and Jenna will win with $2014$ coins.

$\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins.

$\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins.

$\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins.

$\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.

Solution 1

We split into 2 cases: 2013 coins, and 2014 coins.

$\textbf{2013 coins:}$ Notice that when there are $5$ 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 $3$ coins. Then, whenever Barbara takes coins, Jenna will take the amount that makes the total coins taken in that round $5$ . (For instance, if Barbara takes $4$ coins, Jenna will take $1$ ). Eventually, since $2010=0 (\text{mod }5)$ it will be Barbara's move with $5$ 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 $5$ . Since $2013=3 (\text{mod }5)$ , it will be Barbara's move with $3$ coins remaining, so she will have to take $2$ coins, allowing Jenna to take the last coin. Therefore, Jenna will win with $2013$ coins.

$\textbf{2014 coins:}$ If Jenna moves first, she will take $1$ coin, leaving $2013$ coins, and she wins as shown above. If Barbara moves first, she can take $4$ coins, leaving $2010$ . After every move by Jenna, Barbara will then take the number of coins that makes the total taken in that round $5$ . Since $2010=0\text{(mod }5)$ , it will be Jenna's turn with $5$ coins left, so Barbara will win. In this case, whoever moves first wins.

Based on this, the answer is $\boxed{\textbf{(B)}}$

Solution 2

We start with some key observations.

First, notice that when there are $0 (\text{mod }5)$ coins left, whoever moves first loses, since the other player can keep taking exactly $5$ minus the amount the first player took, eventually removing the last coin.

Furthermore, notice that if Barbara is going first when there are $1 (\text{mod }5)$ coins left, she will lose, since the Jenna can follow a similar strategy to the $0 (\text{mod }5)$ case. This will leave Barbara with only 1 coin at the end, in which she loses her turn and Jenna wins.

With these in mind, we split into 2 cases: 2013 coins, and 2014 coins.

$\textbf{2013 coins:}$

If Barbara goes first, she can remove either $2$ or $4$ coins. If she removes $2$ coins, then Jenna will remove $1$ coin, so there will be $0 (\text{mod }5)$ left (hence Barbara will lose). Alternatively, if she removes $4$ coins, then Jenna will remove $3$ coins, so there will be $1 (\text{mod }5)$ left (and Barbara will lose again). In both of these cases, Jenna will win.

If Jenna goes first, she will remove $3$ coins, so there will be $0 (\text{mod }5)$ coins left. Hence Jenna will win if she goes first.

Putting these together, Jenna will always win with 2013 coins.

$\textbf{2014 coins:}$

If Barbara goes first, she will remove $4$ coins, so there will be $0 (\text{mod }5)$ coins left. Hence Barbara will win.

Or, if Jenna goes first, she will remove $3$ coins, so there will be $1 (\text{mod }5)$ coins left. Hence Jenna will win.

Based on these, whoever goes first will win with 2014 coins.

Therefore, the answer is $\boxed{\textbf{(B)}}$ .

~xHypotenuse

Video Solution

https://youtu.be/I6JgSPbL6gY ~Math Problem Solving Skills

2013 AMC 12B (Problems • Answer Key • Resources)
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2013 AMC 12B Problems/Problem 18

Contents

Problem

Solution 1

Solution 2

Video Solution

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