Three rounds of cup swapping

Question

A jug of wine and a jug of water hold the same volume. Round one: pour a cup of wine into the water, then pour a cup of the mixture back into the wine. Round two: the same, a cup from the wine jug into the water jug, then a cup of the new mixture back. Round three: once more. The cups are always full, nothing spills, and you can stir as much or as little as you like at every stage. After three rounds, is there more water in the wine jug or more wine in the water jug? What about after a hundred rounds?

Accepted Answer

Still exactly equal, after three rounds, a hundred rounds, or any number of rounds. Why. Every round moves one cup out of the wine jug and one cup into it, so after each complete round both jugs hold exactly their original volume. Freeze the picture at any such moment. The wine jug contains its original volume, some of which is now water. Every drop of water sitting in the wine jug is occupying space that a drop of wine used to occupy, and that displaced wine has not vanished: the only other place it can be is the water jug. So water-in-wine equals wine-in-water, always. Why the fractions dis…

Three rounds of cup swapping

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