7,8,9,10 |
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Four dwarfs are sitting around a round table. In front of each is a mug of milk. One of the dwarfs pours $1 / 4$ of his milk into the mug of the dwarf to his right. Then the dwarf to his right does the same. Next, the following dwarf to the right does the same, and finally, the fourth dwarf pours $1 / 4$ of the milk that is left in his mug into the first dwarf's mug. Together, there are 2 liters of milk in all the mugs. How much milk was there initially in the mugs if

a) in the end, all dwarfs have the same amount of milk?

b) in the end, all dwarfs have the same amount of milk as they had at the beginning?
a) In the end, all the gnomes ended up with half a liter of milk. This means that before the last pouring, the 4th gnome had $4 / 3.1 / 2 = 2 / 3$ liters, the 1st gnome had $-1 / 2 - 1 / 6 = 1 / 3$ liters, and the other two had half a liter each. Similarly, before the third pouring, the 3rd gnome had $2 / 3$ liters, the 4th gnome had $-2 / 3 - 1 / 6 = 1 / 2$ liters, the 2nd gnome also had half a liter, and the 1st gnome had $-1 / 3$ liters. Therefore, before the second pouring, the 2nd gnome had $2 / 3$ liters, the 3rd and 4th gnomes had half a liter each, and the 1st gnome had $1 / 3$ liters. Finally, at the beginning, the 1st gnome had $4 / 9$ liters, the 2nd gnome had $5 / 9$ liters, and the 3rd and 4th gnomes had half a liter each.

b) Since in the end all the gnomes had as much milk as they started with, each gave as much milk as they received. If the 1st gnome had 4 parts of milk, he poured one part to the 2nd gnome, which should be a quarter of his milk after pouring (so he could pour one part to the 3rd gnome). This means the 2nd gnome had 3 parts. Similarly, the 3rd and 4th gnomes had 3 parts each. Therefore, the ratio of the volume of milk among the gnomes was $4: 3: 3: 3$.

## Answer

a) The 1st gnome had $4 / 9$ liters, the 2nd gnome had $5 / 9$ liters, and the 3rd and 4th gnomes had half a liter each.

b) The 1st gnome had $8 / 13$ liters, and the others had $6 / 13$ liters each.