9.4. Six different natural numbers from 6 to 11 are placed on the faces of a cube. The cube was rolled twice. The first time, the sum of the numbers on the four side faces was 36, and the second time it was 33. What number is written on the face opposite the one with the number 10? Justify your answer.
Solution: The sum of the numbers on all 6 faces of the die is 51. Therefore, the sum of the numbers on the top and bottom faces during the first roll is 15, and during the second roll, it is 18. The number 15 can be obtained (as the sum of two different integers from the interval [6; 11]) in two ways: 15 = 9 + 6 = 8 + 7. The number 18 also in two ways: 18 = 7 + 11 = 8 + 10. If the numbers 7 and 8 lie on opposite faces, then both of the latter representations are impossible. Therefore, of the two specified representations of the number 15, the first one holds, and in the die, the number 9 lies opposite the number 6. Now, if during the second roll, the numbers on the top and bottom faces were 8 and 10, it is clear that the number 10 lies opposite the number 8. And if during the second roll, the numbers on the top and bottom faces were 7 and 11, then on one pair of opposite side faces are the numbers 6 and 9, and on the other - the remaining 8 and 10, and again, the number 10 lies opposite the number 8.

Answer: 8.

Note: The problem can be easily reduced to a regular die as follows: remove 5 dots from each face. Then the die will become regular, and the condition will read: "The first time, the sum of the numbers on the four side faces was 16, the second time - 13. What number is written on the face opposite the one where the number 5 is written?". Further reasoning does not change fundamentally.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct justified answer | 7 points |
| Correct answer, supported by a correct example of a die (any of the two possible) | 3 points |
| Impossibility of some situations is justified (for example, that 10 cannot lie opposite 11 or that the sum during the first roll cannot be realized as 6 + 9 + 10 + 11), but not all impossible situations are analyzed | 2 points |
| The problem is correctly reduced to a regular die |  |
| Answer without justification (or incorrect answer) | 1 point |