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. [Just went to check with colleague Karoly Bezdek to see if this is up-to-date, but he wasn't in. Will supplement if I discover more later.] R. On Thu, 15 Apr 2004, R. William Gosper wrote:
Is r=sqrt 2 the largest disk coverable with four unit disks? --rwg
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