Maximally Even Sets

Richard Krantz, Jack Douthett and John Clough
Bridges: Mathematical Connections in Art, Music, and Science (2000)
Pages 193–200


We wish to seat eight dinner guests--four women and four men--about a round table so that the guests of each gender are distributed as evenly as possible. The obvious solution is to seat men and women alternately. Now suppose we have five women and three men. Ignoring rotations and reflections, there are essentially five ways to seat them (Fig. 1). Which of these is the most (maximally) even distribution? (We will see later that the optimum distribution for one gender guarantees the same for the other.) On the basis of the informal "most even" criterion, the best choice seems to be Fig. Ie (which happens to be the only arrangement that avoids seating three or more women together). This is a relatively simple case, but "dinner table" cases with larger numbers also have a unique best solution. How can we formalize our intuition about evenness for all such cases?