#### Problem Of the week: 24th September 2023

#### Dylan Guo

Alan and Claire live by the old Scottish saying, “Never have whiskey without water, nor water without whiskey!” So one day, when Alan has in front of him a glass of whiskey, and Claire has in front of her a same-sized glass of water, Alan takes a spoonful of his whiskey and puts it in Claire’s water. Claire stirs her whiskey-tinted water and then puts a spoonful of this mixture back into Alan’s whiskey to make sure they have exactly the same amount to drink.

Is there more water in Alan’s whiskey, or more whiskey in Claire’s water? And does it matter how well Claire stirred?

The size and shape of the spoon does not matter.

Think about the composition of each glass after the mixing.

One way to approach this problem quickly is by thinking in extremes. Suppose the spoon was the same size as the entire glass. In that case, putting Alan’s “spoonful” of whisky into Claire’s water would entail mixing both glasses together, leading to a mixture that’s half water and half whisky. Then, when Claire returns a “spoonful” of this mixture to Alan’s glass, there would be exactly half water and half whisky in both glasses.

So in this extreme, there would be the same amount of water in Alan’s whisky as there is whisky in Claire’s water. Indeed, this is the solution no matter the size of the spoon.

To answer more carefully, let’s assume each glass has 100ml of each liquid to start with: Alan’s has 100 mL of whisky and Claire’s has 100 ml of water. Since the liquid transfers consist of removing and adding a spoonful to each glass, the net amount of liquid changed in each glass is zero. Thus, both glasses end with the same amount of liquid they started with 100ml.

This means if Alan has x ml of water in his glass at the end, then he must have exactly 100-x ml of whisky. Since we know that there’s 100 ml of whisky in total, this means there must be x ml of whisky in Claire’s glass.

So the water in Alan’s glass must have displaced whisky in Alan’s glass one-for-one, such that there is exactly the same volume of water in Alan’s glass as there is whiskey in Claire’s glass. This will be the case no matter how well Claire mixed!