243. Seven dwarfs are sitting around a round table. In front of each of them is a mug. Some of these mugs contain milk. One of the dwarfs pours all his milk into the mugs of the others equally. Then his right neighbor does the same. Then the next right neighbor does the same, and so on. After the last, seventh dwarf has poured his milk to everyone else, each mug contains as much milk as it did at the beginning. Together, all the mugs contain 3 liters of milk. How much milk was originally in each mug?
243. Answer: $\frac{6}{7}, \frac{5}{7}, \frac{4}{7}, \frac{3}{7}, \frac{2}{7}, \frac{1}{7}, 0$ liters.

To verify that the given numbers serve as the answer is not difficult: after the first gnome pours the milk (1/7 to each of the others), the distribution remains the same but shifted by one gnome, and the sum $(1+2+3+\ldots+6) / 7$ is exactly 3. It is also necessary to prove that there are no other solutions. Suppose this is not the case.

Let $x$ be the maximum amount of milk that any gnome $\Gamma$ has at any time during the pouring, when it is his turn to pour. Then, after a cycle of 7 "pourings" (which can be repeated indefinitely, so $\Gamma$ can be considered the first in the cycle), $\Gamma$ will have no more than $6 \cdot x / 6 = x$ liters; and equality is possible only if each of the other 6 gnomes pours exactly $x / 6$ liters to $\Gamma$.

Thus, from the condition, it follows that each gnome pours the same amount $x$ of milk and after receiving $\boldsymbol{k}$ portions, he has $k x / 6$ liters in his mug ($k=1,2, \ldots, 6$), and the value of $x$ is determined by the condition that the total amount of milk is 3 liters.

$\nabla$ This problem, which is reminiscent of the charming characters of the Disney film "Snow White and the Seven Dwarfs," was recognized by the readers of "Kvant" as one of the best problems of the year. The answer here is not hard to guess - it is natural to assume that with each pouring, the distribution of milk only shifts by one gnome. However, this is not enough - it is also necessary to prove the uniqueness of the solution (which can be done in many different ways).