I delayed sending this until I'd been able to consult colleague Karoly Bezdek. He doesn't have anything to add. Perhaps the Novotn'y paper hasn't appeared 'cos it turned out to be anticipated by Bezdek? R. In my notes I discover ------ \section{{\Large D3. Covering a circle with equal disks.}} Pavel Novotn\'y, On covering a circle by five and six unit circles, {\it Stud.\ Univ.\ Transport \& Communications \v{Z}ilina}, (to appear). Hans Melissen, Loosest coverings of an equilateral triangle, {\it Math.\ Mag.}, {\bf70}(1997) 118--124. ------- but the 10 `Pavel Novotny' hits that I got on MathSciNet didn't include this. Melissen would be a good person to ask about this problem. R. On Thu, 15 Apr 2004, Richard Guy wrote:
Here is UPIG D3:
Covering a circle with equal disks. The problem of completely covering a circular region, by placing overit, one at a time, five smaller equal circular disks was familiar to frequenters of English fairs a century ago. It can be done if the radius of the smaller disks exceed $0.609383\ldots$ of that of the circular region. For a discussion of Neville's solution, see Rouse Ball (though the number is given incorrectly in some editions). What is the minimum radius for coverings by other numberings of equal disks? The cases of three, four and seven disks are easy, and Bezdek gives solutions for five and six disks. For conjectured extremals for other numbers of disks up to 20, see Zahn.
K.~Bezdek, \"Uber einige optimale Konfigurationen von Kreisen, {\it Ann.\ Univ.\ Sci.\ Budapest.\ E\"otv\"os Sect.\ Math.}, {\bf27}(1984) 143--151; {\it MR} {bf87f}:52020.
J.~Moln\'ar, \"Uber eine elementargeometrische Extremalaufgabe, {\it Mat.\ Fiz.\ Lapok}, {\bf49}(1942) 249--253; {\it MR} {\bf8}, 218.
E.~H.~Neville, Solutions of numerical functional equations, {\it Proc.\ London Math.\ Soc.}(2), {\bf14}(1915) 308--326.
W.~W.~Rouse Ball, {\it Mathematical Recreations \& Essays}, 10th ed., Macmillan, New York, 1931, 253--255; 12th ed., Univ.\ of Toronto Press, Toronto, 1974, 97--99.
C.~T.~Zahn, Black box maximization of circular coverage, {\it J.\ Res.\ Nat.\ Bur.\ Stand.\ B}, {\bf66}(1962) 181--216.
On Thu, 15 Apr 2004, R. William Gosper wrote:
Is r=sqrt 2 the largest disk coverable with four unit disks? --rwg