Home > Challenge > Coffee, Cream, and Making a Wish

## Coffee, Cream, and Making a Wish

We have looked at several solutions to the classic Coffee and Cream mixture problem:

Suppose you have a cup of coffee and a cup of cream.  If you take a spoonful of cream, mix it up with the coffee, then take a spoonful of that mixture and add it back to the cream, is there more cream in the coffee, or more coffee in the cream?

I offered a solution that begins by making a wish.  That is, I wished that each cup initially contained 10 spoonfuls of liquid; then I proceed to solve the problem algebraically, by following spoonfuls of liquid back and forth between cups.  To illustrate the real power of the make-a-wish strategy, though, let’s wish that each cup started with one spoonful of liquid.

In this case, the act of transferring one spoonful of cream into the coffee amounts to pouring all the cream in the coffee.  Now, mix it up.  The mixture on the left is obviously half coffee and half cream.

Now, pour a spoonful of the mixture, or half, back into the empty cup.

The two cups now contain exactly the same solution:  half coffee, and half cream!  So the answer to the question is obviously that there are equal amounts of cream in the coffee and coffee in the cream!

Be sure to check out the elegant solution to the problem, as well!