Dear Uncle Colin,

The puzzle asks: you have two glasses with equal volumes of water and juice. You take a tablespoon of water and mix it into the juice; you take a tablespoon of the juice-water mix and mix it into the water. Is there more water in the juice, or juice in the water? Apparently the amounts are the same, but I don’t see how they can be!

Combining Liquids And Stirring Strangely - It’s Confusing!

Hi, CLASSIC, and thanks for your message!

There are many ways to approach this puzzle, and I’m going to leave what I think is the simplest one until last.

Algebra

Initially, one glass contains $V$ of water; the other contains $V$ of juice.

Take a volume $v$ of water and add it to the juice; there is now $V - v$ of water in one glass; in the other, there is $V$ of juice and $v$ of water.

Now the tricky bit: we’re going to move a total of $v$ of the mixture back into the first glass, but it could be all juice, all water, or any mixture in between. So, let’s say the $v$ is made up of $t$ of juice and $v-t$ of water.

After moving it, the first glass has $(V - v) + (v-t)$, or $V-t$ of water and $t$ of juice.

The second glass has $V-t$, of juice, and $v - (v-t)$ or $t$ of water.

There’s as much juice in the water as there is water in the juice.

Cards

The algebra is all well and good, but it’s not exactly intuitive. Instead, I like to think about things you can count, like cards of different colours.

Imagine you have a number of red cards and a number of blue cards. You take a certain number of red cards and mix them with the blue ones; you take the same number from the mixed pack and put them back in the red pack.

Now, there’s the same number of cards in each pack, and there’s still the same number of red and blue cards split between them. For every red card in the blue pack, a blue card must have gone to the red pack, and vice versa: there are as many red cards in the blue pack as vice versa.

Intuitively

Everything in the juice glass that isn’t juice, is water.

Everything in the water glass that isn’t water, is juice.

There’s the same amount of water and juice altogether, so the “isn’ts” must be the same.

Hope that helps!

- Uncle Colin