Zur Zerlegung von Kreisüberdeckungen \(\ddot{\mathfrak U}(r,1)\) der Ebene in zwei Kreispackungen \(\mathcal P(r,1)\) und \(\mathcal Q(r)\). (On the decomposition of circle packings \(\ddot{\mathfrak U}(r,1)\) of the plane into two circle packings \(\mathcal P(r,1)\) and \(\mathcal Q(r)\)) (Q1823470)

The covering (of the title) is that of the Euclidean plane by circular discs of radii r and 1 (some of each radius, and \(0<r<1)\) in which every disc covers a point of the plane not covered by any of the other discs. The author proves that such a covering can be decomposed into two packings - one by discs of radii r and 1, the other by discs and radius 1 - if and only if \(r=\sqrt{2}-1\). Furthermore, there is just one such covering, a picture of which is given. For related and similar problems see: \textit{J. Pach}, Diskrete Geometrie, 2. Kolloq., Inst. Math. Univ. Salzburg 1980, 169-178 (1980; Zbl 0443.52017); the author, Ann. Univ. Sci. Budapest Rolando Eötvös, Sect. Math. 30, 25-33 (1987; Zbl 0641.52007); \textit{K.Bezdek} and the author, to appear in Beitr. Algebra Geom. 28.