A die is a cube with its faces numbered 1 through 6 . One red die and one blue die are rolled. The sum of the numbers on the top two faces is determined. What is the probability that this sum is a perfect square?
When the red die is rolled, there are 6 equally likely outcomes. Similarly, when the blue die is rolled, there are 6 equally likely outcomes.

Therefore, when the two dice are rolled, there are $6 \times 6=36$ equally likely outcomes for the combination of the numbers on the top face of each. (These outcomes are Red 1 and Blue 1, Red 1 and Blue 2, Red 1 and Blue 3,.., Red 6 and Blue 6.)

The chart below shows these possibilities along with the sum of the numbers in each case:

| Blue Die |  |  |  |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  |  |  |  |  |  |
| Red | 1 | 2 | 3 | 4 | 5 | 6 |  |
|  | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|  | 4 | 4 | 5 | 6 | 7 | 8 | 9 |
|  | 5 | 6 | 6 | 7 | 8 | 9 | 10 |
| 6 | 7 | 8 | 9 | 9 | 10 | 11 |  |
|  |  |  | 10 | 11 | 12 |  |  |

Since the only perfect squares between 2 and 12 are 4 (which equals $2^{2}$ ) and 9 (which equals $3^{2}$ ), then 7 of the 36 possible outcomes are perfect squares.

Since each entry in the table is equally likely, then the probability that the sum is a perfect square is $\frac{7}{36}$.

ANSWER: $\frac{7}{36}$

#