On Partitions of an Equilateral Triangle

Let T denote a closed unit equilateral triangle. For a fixed integer n, let d n denote the infimum of all those x for which it is possible to partition T into n subsets, each subset having a diameter not exceeding x. We recall that the diameter of a plane set A is given by where ρ (a, b) is the Euclidean distance between a and b. In this note we determined d n for some small values of n. Typical values of d n are given in Table I. These values were obtained by three methods. As would be expected, as the value of n increases, the complexity of the argument needed to obtain d n also increases. We begin with the simplest case.