Some years ago I was invited to take part in a fund-raising competition, [1]. The rules had an interesting feature that led me to analyse the situation mathematically. This resulted in some interesting formulae and one inequality, which I will describe here.

The competition offered a single prize and the entry form looked similar to the partially-completed example illustrated below.

The form listed 40 categories similar to the ones shown. Participants were required to write down an ‘answer’ for each category, e.g. for ‘an animal’, you might choose DOG. No other indications were given, so each answer was a completely free choice.