Sources page biographical material

5.F.2. OTHER HAMILTONIAN CIRCUITS
For circuits on the n cube, see also 5.F.4 and 7.M.1,2,3.

For circuits on the chessboard, see also 6.AK.

Le Nôtre. Le Labyrinte de Versailles, c1675. This was a hedge or garden maze, but the objective was to visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at least one fountain. Some fountains were not at path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian problem, except that one fountain was located at the end of a short dead end. [Fisher, op. cit. in 5.E.1, pp. 49, 79, 130 & 144-145, with contemporary diagram on p. 144. He says there are 39 fountains, but the diagram has 40.]
T. P. Kirkman. On the partitions of the R pyramid, being the first class of R gonous X edra. Philos. Trans. Roy. Soc. 148 (1858) 145 161.

W. R. Hamilton. The Icosian Game. 4pp instructions for the board game. J. Jaques and Son, London, 1859. (Reproduced in BLW, pp. 32-35, with frontispiece photo of the board at the Royal Irish Academy.)

For a long time, the only known example of the game, produced by Jaques, was at the Royal Irish Academy in Dublin. This example is inscribed on the back as a present from Hamilton to his friend, J. T. Graves. It is complete, with pegs and instructions. None of the obvious museums have an example. Diligent searching in the antique trade failed to turn up an example in twenty years, but in Feb 1996, James Dalgety found and acquired an example of the board -- sadly the pegs and instructions were lacking. Dalgety obtained another board in 1998, again without the pegs and instructions, but in 1999 he obtained another example, with the pegs.

Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53-54 & 102; 1917: 310, pp. 49 & 97. The garden of a French palace has a maze with 31 points to see. Find a path past all of them with no repeated edges and no crossings. The pattern is clearly based on the Versailles maze of c1675 mentioned above, but I don't recall the additional feature of no crossings occurring before.

T. P. Kirkman. Solution of problem 6610, proposed by himself in verse. Math. Quest. Educ. Times 35 (1881) 112 116. On p. 115, he says Hamilton told him, upon occasion of Hamilton presenting him 'with his handsomest copy of the puzzle', that Hamilton got the idea for the Icosian Game from p. 160 of Kirkman's 1858 article,

Lucas. RM2, 1883, pp. 208 210. First? mention of the solid version. The 2nd ed., 1893, has a footnote referring to Kirkman, 1858.

John Jaques & Son. The Traveller's Dodecahedron; or, A Voyage Round the World. A New Puzzle. "This amusing puzzle, exercising considerable skill in its solution, forms a popular illustration of Sir William Hamilton's Icosian Game. A wood dodecahedron with the base pentagon stretched so that when it sits on the base, all vertices are visible. With ivory? pegs at the vertices, a handle that screws into the base, a string with rings at the ends and one page of instructions, all in a box. No date. The only known example was obtained by James Dalgety in 2002.

Pearson. 1907. Part III, no. 60: The open door, pp. 60 & 130. Prisoner in one corner of an 8 x 8 array is allowed to exit from from the other corner provided he visits every cell once. This requires him to enter and leave a cell by the same door.

Ahrens. Mathematische Spiele. 2nd ed., Teubner, Leipzig, 1911. P. 44, note, says that a Dodekaederspiel is available from Firma Paul Joschkowitz -- Magdeburg for .65 mark. This is not in the 1st ed. of 1907 and the whole Chapter is dropped in the 3rd ed. of 1916 and the later editions.

Anonymous. The problems drive. Eureka 12 (Oct 1949) 7-8 & 15. No. 3. How many Hamiltonian circuits are there on a cube, starting from a given point? Reflections and reversals count as different tours. Answer is 12, but this assumes also that rotations are different. See Singmaster, 1975, for careful definitions of how to count. There are 96 labelled circuits, of which 12 start at a given vertex. But if one takes all the 48 symmetries of the cube as equivalences (six of which fix the given vertex), there are just 2 circuits from a given starting point. However, these are actually the same circuit started at different points. Presumably Kirkman and Hamilton knew of this.

C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed., Gollancz, London, 1971, pp. 24-25. Roman knobbed dodecahedra -- an ancient solid version??

R. E. Ingram. Appendix 2: The Icosian Calculus. In: The Mathematical Papers of Sir William Rowan Hamilton. Vol. III: Algebra. Ed. by H. Halberstam & R. E. Ingram. CUP, 1967, pp. 645 647. [Halberstam told me that this Appendix is due to Ingram.] Discusses the method and asserts that the tetrahedron, cube and dodecahedron have only one unlabelled circuit, the octahedron has two and the icosahedron has 17.

David Singmaster. Hamiltonian circuits on the regular polyhedra. Notices Amer. Math. Soc. 20 (1973) A 476, no. 73T A199. Confirms Ingram's results and gives the number of labelled circuits.

David Singmaster. Op. cit. in 5.F.1. 1975. Carefully defines labelled and unlabelled circuits. Discusses results on regular polyhedra in 3 and higher dimensions.

David Singmaster. Hamiltonian circuits on the n dimensional octahedron. J. Combinatorial Theory (B) 18 (1975) 1 4. Obtains an explicit formula for the number of labelled circuits on the n dimensional octahedron and shows it is  (2n)!/e. Gives numbers for n  8. In unpublished work, it is shown that the number of unlabelled circuits is asymptotic to the above divided by n!2ⁿ4n.

Angus Lavery. The Puzzle Box. G&P 2 (May 1994) 34-35. Alternative solitaire, p. 34. Asks for a knight's tour on the 33-hole solitaire board. Says he hasn't been able to do it and offers a prize for a solution. In Solutions, G&P 3 (Jun 1994) 44, he says it cannot be done and the proof will be given in a future issue, but I never saw it.