Kustarev A.A.

The faces of a cube are numbered from 1 to 6. The cube was rolled twice. The first time, the sum of the numbers on the four side faces was 12, the second time it was 15. What number is written on the face opposite the one where the digit 3 is written?
The sum of the numbers on all faces of the cube is 21.

## Solution

Notice that the sum of all numbers written on the cube is 21. The sum of the numbers on the top and bottom faces in the first and second cases is 9 and 6, respectively.

After the first roll, it is clear that either 3 is opposite 6, or 4 is opposite 5. Suppose 4 is opposite 5. But after the second roll, it is clear that either 1 is opposite 5, or 2 is opposite 4. This is a contradiction, so 3 is opposite 6.

## Answer

6.