10. A game consists of 10 rolls of a normal cubic die with faces numbered from 1 to 6. At the end, the scores obtained are summed, with the rule that if a 6 is rolled, the points of the next roll are counted doubled, and if a 6 is rolled on the last roll, the player gets one (only one) additional roll to add to the score (not doubled). Which of these final scores cannot be obtained?
(A) 64
(B) 78
(C) 92
(D) 114
(E) 120.

## Numerical answer problems - 8 points
10. The answer is (D). Indeed, suppose by contradiction that 114 is an admissible score. Let $N$ be the number of throws with which it was obtained; $N$ can be 10 or 11. The first score is less than or equal to six, so at least $114-6=108$ was obtained with $N-1$ throws; note that $108=12 \times 9$, so if $N=10$, the only possible sequence of scores would be a 6 and nine consecutive 12s, but then the last throw would have given 6 and there would have been an eleventh throw.

Therefore, $N=11$. It follows that at least 108 was obtained with 10 throws (from the second to the eleventh). Note that the eleventh throw gives a score less than or equal to six, so at least 102 must have been obtained with 9 throws. Suppose that the scores of these throws are not all 12. Then there is a score less than or equal to 10 (11 cannot be obtained). Suppose this happened on the tenth throw, which consequently must be a 6. This is impossible because at least $96=8 \times 12$ would have been obtained with 8 throws (from the second to the ninth), which should all give 12, but then the tenth would give 12 and not 6. Therefore, the score less than or equal to 10 is obtained before the tenth throw, and the score on the next throw is less than or equal to 6; subtracting these two scores, it follows that at least $102-16=86$ was obtained with only seven throws, and this is impossible.

In conclusion, the only admissible sequence is the one given by an initial 6, nine consecutive 12s, plus the score $p$ of the eleventh throw, and the total score is $114+p$ which cannot coincide with 114 because $p$ is between 1 and 6. We have reached a contradiction.