Water and Wine

Some time ago I heard an interesting riddle that gave out quite a few headaches. I've been told this puzzle originated from The Republic by Plato, though I don't know if that's the real source of this riddle. His congratulations go out to whoever can solve this puzzle by thought experiment alone.

Here is the riddle:
1. Get 1 cup of wine and 1 cup of water.
2. Take a table spoon of wine and put it into the water.
3. Mix thoroughly.
4. Take a table spoon of the now water and wine mixture and put it back into the wine.

Now the question: at the end of the procedure which cup has a higher ratio of foreign substance to original substance? Is the water cup more polluted by wine or the wine cup more polluted by the water?

Most people's first reaction would be to say the water cup has surely been more polluted. We put a table spoon of pure wine into the water, but we only put a mixture of water and wine back into the wine glass. This is the most obvious observation of course, and there is a subtle observation that might lean you towards the other direction.

Even though we put a table spoon of pure wine into a the glass of water, and we put the mixture into the glass of wine. The effect of the wine glass being less than full means whatever we put into it should contaminate it more than if the glass had been full.

The last paragraph gives the argument for why it may be either side which is more polluted, or maybe an argument why they might be the same. I suggest to take your side now, as the rest of the post will describe the solution.

There are lots of ways to solve this puzzle, I will give the way I find the most elegant. It is similar to a type of argument that's used in mathematics called a combinatorial argument. We measure the same thing in two different ways and then since both measurements are of the same thing the measurements must be the same. This argument has the advantage in this situation of not worrying about the sequence of events outlined in the procedure. If we were to simply work out the amount of wine and water in each cup at each stage then we would come to a mess of fractions.

In order to use the strategy it's necessary to name the different pieces of fluid. The following image shows how I chose to label the water and wine in each glass. Now consider ways we can make a full glass of fluid. If we look at what's in the water glass it must measure up to 1 cup. This is because we took 1 table spoon of fluid out and put back 1 table spoon. Therefore $$A + B = 1 cup$$. On the other hand we started with 1 cup of water, so if we add up all the water we should get 1 cup. This means $$A + D = 1 cup.$$ These two equations cannot both be true unless $B = D$. So we can conclude both cups are equally polluted.

I hope you all enjoyed this one :).