Snow White gave each dwarf the same number of marbles. The dwarfs rolled a die. Whatever number of dots fell, that many marbles they had to return to Snow White. Each of the six dwarfs rolled a different number of dots. How many dots did the seventh dwarf roll if the dwarfs were left with 46 marbles in total? How many marbles did Snow White give each dwarf at the beginning?

(L. Hozová)
Snow White gave each dwarf $x$ balls,

the dwarfs had a total of $7 x$ balls.

During the throws of six dwarfs, $(1+2+3+4+5+6)$ dots fell, a total of 21 dots.

So the dwarfs gave Snow White 21 balls and had $(7 x-21)$ balls left.

The seventh dwarf rolled $y$ dots and gave Snow White $y$ balls. The dwarfs had $(7 x-21-y)$ balls left in total.

It holds that:

$$
7 x-21-y=46 \text {, }
$$

thus

$$
7 x-y=67
$$

from this relation it is clear that $7 x>67$, i.e., $x>9$.

If $x=10$, then $7 x=70$, then $y=70-67=3$.

If $x=11$, then $7 x=77$, then $y=77-67=10$, but this cannot happen because the highest number of dots on a die is 6. Therefore, no further solutions are possible.

Snow White initially gave each dwarf 10 balls. The seventh dwarf rolled 3 dots.