8.5. The faces of a cube are labeled with six different numbers from 6 to 11. The cube was rolled twice. The first time, the sum of the numbers on the four side faces was 36, the second time - 33. What number is written on the face opposite the one where the digit $10$ is written?
Solution: The sum of the numbers on all faces is

$$
6+7+8+9+10+11=51
$$

On the first roll, the sum of the numbers on the top and bottom faces is 51 $-36=15$, on the second roll $-51-33=18$. Therefore, the sum on the third pair of opposite faces is $51-15-18=18$. The sum of 18 can be obtained in two ways: $11+7$ or $10+8$. Therefore, on the pairs of faces with a sum of 18, 7 is opposite 11, and 8 is opposite 10.

## Answer: 8