4. 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 face of each die is determined. What is the probability that this sum is a perfect square?
4. 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
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline & & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline & 1 & 2 & 3 & 4 & 5 & 6 & \\
\hline & 2 & 3 & 4 & 5 & 6 & 7 & \\
\hline Red & 3 & 4 & 5 & 6 & 7 & 8 & \\
\hline ie & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline & 0 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\end{tabular}

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: $\frac{7}{36}$