Problem 4.8. Vasya has six identical dice, on the faces of each of which the numbers from 1 to 6 are written (each one only once). Vasya rolled all six dice six times in a row. No number appeared twice on the same die.

It is known that the sum of the numbers on the top faces was 21 on the first roll, and 19, 20, 18, and 25 on the next four rolls. What was the sum on the sixth roll?
Answer: 23.

Solution. Since in six throws, no number appeared twice on any of the dice, each die showed each of the numbers from 1 to 6 exactly once.

Let's calculate the total sum of all numbers that appeared on all dice over the six throws. For one die, this sum is $1+2+3+4+5+6=21$, and for six dice, it is $6 \cdot 21=126$. It remains to calculate the sum of the numbers on the sixth throw: $126-21-19-20-18-25=23 . \quad \square$ Remark. One of the possible configurations of the dice throws is shown below (the first number in the sum is the number on the first die, the second number is on the second, ..., the sixth number is on the sixth).

$$
\begin{aligned}
& 1+5+6+3+5+1=21 \\
& 2+4+3+4+1+5=19 \\
& 3+1+5+2+3+6=20 \\
& 4+2+2+5+2+3=18 \\
& 5+3+1+6+6+4=25 \\
& 6+6+4+1+4+2=23
\end{aligned}
$$