1230.56 – Take your Pills

Suppose that you take two pills every day, call them A and B. One day one A pill fell out of its bottle, and, because you're in a hurry, two B pills fell out of their bottle. The three pills landed on the table and they got mixed up. You can't tell them apart by looking, so you have three pills that look alike and you need to take one A and one B. How can you figure out which two pills to take? (You can't throw the three pills away and start over—they're expensive.)


Take out another A pill and put it with the three you already have. Now you have two of each and you can't tell which are which (except for the one A pill you just took out). Take a pill splitter; it works really well. Take a pill, any pill, split it in half, and put one half to your left on the table and the other one to your right. Do this with each of the other three pills, each time putting one half on the left and one half on the right. You will wind up with two little piles of half-pills, with four half-pills in each pile. You still don't know which of these halves are A's and which are B's. But it doesn't matter! If you stop to think for a minute, you will see that each pile contains two half-A's and two half-B's. Take the left-hand pile for today's pills and save the right-hand pile for tomorrow.

This fascinating problem appears in Puzzles to Unravel the Universe, by Harvard Professor Cumrun Vafa, p. 147