Covering a rectangle with six and seven circles (Q1962064)

Finitely many circles (circular discs) cover a rectangle if every point of the rectangle is in one (or more) of the circles. A covering of a \(1\times b\) rectangle by \(n\) circles is thinnest if the (common) radius is smallest (among all coverings by \(n\) circles). \textit{A. Heppes} and \textit{H. Melissen} [Period. Math. Hung. 34, No. 1-2, 65-81 (1997; Zbl 0880.52008)] determined the thinnest covering by up to 5 circles (for all \(b\geq 1)\) and by 7 circles for some \(b\), namely \(1\leq b\leq 1.34457\dots\) and \(b>3.43017.\dots\) In the paper under review the authors present coverings by 7 circles for the gap (i.e., \(1.34457\dots < b<3.43017\dots)\) which they conjecture to be thinnest, and they determine thinnest coverings by 6 circlcs for \(b\geq 3.11803.\dots\).