Water and Wine

The Setup: One has two barrels, one holding wine and the other an equal volume of water. A cup of wine is taken from the wine barrel and added to the water. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal.

The Problem: which of the two mixtures is purer?

• The problem can be solved without resorting to computation.
• The volume of the cup is irrelevant, as is any stirring of the mixtures.
• Any number of transfers can be made, as long as the volume of liquid in each barrel is the same at the end.

The Solution: The mixtures will be of equal purity. Below is a breakdown of the numbers, presuming 80 liters of fluid per barrel.

Wine Barrel	Action	Water Barrel
80 (all wine)		80 (all water)
	20 (all wine) →
60 (all wine)		100 (80 water, 20 wine)
	← 20 (16 water, 4 wine)
80 (64 wine, 16 water)		80 (64 water, 16 wine)

The easiest way to explain this problem is, that if the volumes of liquid return to the exact same amounts, then, after the transfer of the wine and water, the wine that was transferred to the water has to be the same amount of water that was transferred to the wine.