Ann, Bill and Carol each have some tokens. There are four exchanges of tokens that happen in the order below:

- First, Ann gives Bill the number of tokens that Bill has.
- Next, Ann gives Carol twice the number of tokens that Carol has.
- Next, Bill gives Ann the number of tokens that Ann has.
- Finally, Bill gives Carol twice the number of tokens that Carol has.

After these four exchanges of tokens, Ann, Bill, and Carol each have 36 tokens. How many tokens did Ann have before any were exchanged?
Solution 1

Suppose Ann, Bill, and Carol begin with $A, B$, and $C$ tokens, respectively. The table below shows how many tokens they each have after each exchange:

|  | Ann | Bill | Carol |
| :---: | :---: | :---: | :---: |
| Initially | $A$ | $B$ | $C$ |
| After first exchange | $A-B$ | $2 B$ | $C$ |
| After second exchange | $A-B-2 C$ | $2 B$ | $3 C$ |
| After third exchange | $2(A-B-2 C)$ | $2 B-(A-B-2 C)$ | $3 C$ |
|  | $2 A-2 B-4 C$ | $-A+3 B+2 C$ | $3 C$ |
| After fourth exchange | $2 A-2 B-4 C$ | $-A+3 B+2 C-6 C$ | $3 C+6 C$ |
|  | $2 A-2 B-4 C$ | $-A+3 B-4 C$ | $9 C$ |

We know that after the four exchanges are complete, Ann, Bill, and Carol each have 36 tokens. Therefore, $9 C=36$, so $C=4$. Substituting this into $2 A-2 B-4 C=36$ and $-A+3 B-4 C=36$ gives $2 A-2 B=52$ and $-A+3 B=52$.

Dividing the equation $2 A-2 B=52$ by 2 gives $A-B=26$ and adding this to $-A+3 B=52$ gives $2 B=78$ so $B=39$.

Substituting this into $A-B=26$ gives $A=65$.

## Solution 2

The number of tokens that Ann has before the fourth exchange is the same as the number she has after the exchange.

During the exchange, the number of tokens that Carol has triples, so before the fourth exchange,

Carol had $\frac{36}{3}=12$, which means Bill gives Carol $36-12=24$ tokens in the fourth exchange. Thus, in the fourth exchange, the number of tokens that Ann has does not change, the number of tokens that Bill has decreases by 24, and the number of tokens that Carol has increases by 24 .

By similar reasoning, during the third exchange, the number of tokens that Carol has does not change, the number of tokens that Ann has increases by 18, and the number of tokens that Bill has decreases by 18 .

During the second exchange, the number of tokens that Bill has does not change, the number of tokens that Carol has increases by 8, and the number of tokens that Ann has decreases by 8. During the fist exchange, the number of tokens that Carol has does not change, the number of tokens that Bill has increases by 39, and the number of tokens that Ann has decreases by 39 . These results are summarized in the table below:

| Exchanges | Ann |  | Bill |  | Carol |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | Before | After | Before | After | Before | After |
| Fourth | 36 | 36 | 60 | 36 | 12 | 36 |
| Third | 18 | 36 | 78 | 60 | 12 | 12 |
| Second | 26 | 18 | 78 | 78 | 4 | 12 |
| First | 65 | 26 | 39 | 78 | 4 | 4 |

Before any tokens are exchanged, Ann has 65 tokens.

ANSWER: 65