1. Introduction

What is the maximum radius rn of a circular disk which can be covered by n closed unit circles? The determination of rn for n ≤ 4 is an easy task: we have and. The problem of finding r5 has been motivated by a game popular on fairs around the turn of the twentieth century [Neville 1915; Ball and Coxeter 1987, pages 97-99]. The goal of the game was to cover a circular space painted on a cloth by five smaller circles equal to each other. The difficulty consisted in the restriction that an “on-line algorithm” had to be used, that is no circle was allowed to be moved once it had been placed. Neville [1915] conjectured that r5 = 1.64100446… and this has been verified by K. Bezdek [1979; 1983] who also determined the value of r6 = 1.7988… . The proofs of these cases are complicated. The case n = 7 is again easy. We have r7 = 2 and if 7 unit circles cover a circle C7 of radius 2, then one of them is concentric with C7 while the centers of the other circles lie in the vertices of a regular hexagon of side y/3 concentric with C7. In his thesis Denes Nagy [1975] claimed without proof that for n = 8 and n = 9 and that, as for n = 7, the best arrangement has (n — l)-fold rotational symmetry.