Recently I came across L. A. Graham’s delightful book of mathematical puzzles [1] for the first time and found there the problem I shall give in a moment. It occurred to me to try to solve it in a more general form.

I was able to find an unexpectedly neat and satisfying answer, using a famous result of E. Helly. The solution provides an excellent example of the type of use to which Helly’s theorem can be put, and a striking example of its power.