. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. IX] UNICURSAL PROBLEMS 189 Using these formulae we can find successively the values of A1,Ai,.,., and £X) .B,,,.... The values of An when n = 2, 3,4, 5, 6, 7, are 2, 4, 9, 20, 48, 115; and of Bn are 1, 2, 5, 12/33, 90 I turn next to consider some problems where it is desired to find a route which will pass once and only once through each node of a given geometrical figure. This is the reciprocal of the problem treated in the first part of this chapter, and
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. IX] UNICURSAL PROBLEMS 189 Using these formulae we can find successively the values of A1,Ai,.,., and £X) .B,,,.... The values of An when n = 2, 3,4, 5, 6, 7, are 2, 4, 9, 20, 48, 115; and of Bn are 1, 2, 5, 12/33, 90 I turn next to consider some problems where it is desired to find a route which will pass once and only once through each node of a given geometrical figure. This is the reciprocal of the problem treated in the first part of this chapter, and is a far more difficult question. I am not aware that the general theory has been considered by mathematicians, though two special cases—namely, the Hamiltonian (or Icosian) Game and the Knight's Path on a Chess-Board—have been treated in some detail. The Hamiltonian Game. The Hamiltonian Game consists in the determination of a route along the edges of a regular dodecahedron which will pass once and only once through every angular point. Sir William Hamilton*, who invented this game—if game is the right term for it—denoted the twenty angular points on the solid by letters which stand for various towns. The thirty edges constitute the only possible paths. The inconvenience of using a solid is considerable, and the dodecahedron may be represented conveniently in. perspective by a flat board marked as shown in the first of the annexed diagrams. The second and third diagrams will answer our purpose equally well and are easier to draw. * See Quarterly Journal of Mathematics, London, 1862, vol. v, p. 305; or Philosophical Magazine, January, 1884, series 5, vol. xvn, p. 42 j also Lncas, vol. II, part Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Ball, W. W. Rouse (Walter William Rouse), 1850-1925.
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