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6.BQ. COVERING A DISC WITH DISCS
The general problem is too complex to be considered recreational. Here I will mainly deal with the carnival version where one tries to cover a circular spot with five discs. In practice, this is usually rigged by stretching the cloth.
Eric H. Neville. On the solution of numerical functional equations, illustrated by an account of a popular puzzle and of its solution. Proc. London Math. Soc. (2) 14 (1915) 308-326. Obtains several possible configurations, but says "actual trial is sufficient to convince" that one is clearly the best, namely the elongated pentagon with 2-fold symmetry. This leads to four trigonometric equations in four unknown angles which theoretically could be solved, but are difficult to solve even numerically. He develops a modification of Newton's method and applies it to the problem, obtaining the maximal ratio of spot radius to disc radius as 1.64091. Described by Gardner and Singleton.

Ball. MRE. 10th ed., 1922. Pp. 253-255: The five disc problem. Sketches Neville's results.

Will Blyth. More Paper Magic. C. Arthur Pearson, London, 1923. Cover the spot, pp. 66-67. "This old "fun of the fair" game has been the means of drawing many pennies from the pockets of frequenters of fairs." Says the best approach is an elongated pentagon which has only 2-fold symmetry.

William Fitch Cheney Jr, proposer; editorial comment. Problem E14. AMM 39 (1932) 606 & 42 (1935) 622. Poses the problem. Editor says no solution of this, or its equivalent, prob. 3574, was received, but cites Neville.

J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Cover the spot, pp. 132-134. Uses discs of diameter 1 5/8 in to cover a circle of diameter 2 1/2 in. This is a ratio of 20/13 = 1.538..., which should be fairly easy to cover?? His first disc has its edge passing through the centre of the circle. His covering pattern has bilateral symmetry, though the order of placing the last two discs seems backward.

Walter B. Gibson. The Bunco Book. (1946); reprinted by Citadel Press (Lyle Stuart Inc.), Secaucus, New Jersey, 1986. Spotting the spot, pp. 24-25. Also repeated in summary form, with some extra observations in: Open season on chumps, pp. 97-106, esp. pp. 102 103. The circles have diameter 5" and the discs "are slightly more than three inches in diameter." He assumes the covering works exactly when the discs have five-fold symmetry -- which implies the discs are 3.090... inches in diameter -- but that the operator stretches the cloth so the spot is unsymmetric and the player can hardly ever cover it -- though it still can be done if one plays a disc to cover the bulge. "There is scarcely one chance in a hundred that the spectator will start correctly ...." On p. 103, he adds that "in these progressive times" the bulge can be made in any direction and that shills are often employed to show that it can be done, though it is still difficult and the operator generally ignores small uncovered bits in the shills' play in order to make the game seem easy.

Martin Gardner. SA (Apr 1959)?? c= 2nd Book, chap. 13. Describes the five disc version as Spot-the-Spot. Cites Neville.

Colin R. J. Singleton. Letter: A carnival game -- covering disks with smaller disks. JRM 24:3 (1992) 185-186. Responding to a comment in JRM 24:1, he points out that the optimum placing of five discs does not have pentagonal symmetry but only bilateral. Five discs of radius 1 can then cover a disc of radius 1.642.., rather than 1.618..., which occurs when there is pentagonal symmetry. He cites Gardner and E. H. Neville. His 1.642 arises because Gardner had truncated the reciprocal ratio to three places.