The Mixing Problem

Say you have two barrels, one containing wine and the other containing an equal amount of water. You take a cup of wine from the wine barrel and add it to the water. If you want, you can stir it some — or not, it doesn't matter. Then take a cup of that wine/water mixture and return it into the wine barrel. The cup should hold the exact same amount so that both barrels once again contain equal volumes (but now mixed). The question is: which one of the mixtures is purer?

Thought about it? The answer is that both mixtures are equally pure! (← click to spoil)

Solution

Okay, this is easier to explain with some pictures. And to be able to picture it, the liquids need to be divided into some unit that can be drawn and counted. You could think of these as drops, milliliters, molecules.... I'll go with ‘parts’. I'll also use wine glasses instead of barrels.

1. To start off, the left glass contains 30 parts of wine and the right glass contains 30 parts of water.

Take a spoonful of wine from the left glass and deposit it into the right glass. I'll now define ‘a spoonful’ as 8 parts.

2. This is an example of what could be the result. The 8 parts of wine have mixed with the water in some random matter. More so if you stirred.

Now transfer a spoonful from the right glass to the left glass. This is exactly 8 parts once again. The amount of those that are wine parts and are water parts is random.

3. Let's say the spoon contained 2 parts of wine and 6 parts of water. Then this is what it could look like now. There's 24 parts of the original liquid in each glass. Thus they are of equal purity!

And it doesn't matter how you randomize those numbers. As long as the volumes transferred are exactly equal, they will always end up equally pure. Down to the last molecule.

To clarify: you can't just lose some liquid. If there's 10 liters of wine missing from the wine barrel, it has to be in the other (water) barrel. And the 10 liters of water that would have normally been in its place has to be in the wine barrel again!