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.

5.F.3. KNIGHT'S TOURS IN HIGHER DIMENSIONS
A. T. Vandermonde. Remarques sur les problèmes de situation. Hist. de l'Acad. des Sci. avec les Mémoires (Paris) (1771 (1774)) Mémoires: pp. 566 574 & Plates I & II. ??NYS. First? mention of cubical problem. (Not given in BLW excerpt.)

F. Maack. Mitt. über Raumschak. 1909, No. 2, p. 31. ??NYS -- cited by Gibbins, below. Knight's tour on 4 x 4 x 4 board.

Dudeney. AM. 1917. Prob. 340: The cubic knight's tour, pp. 103 & 229. Says Vandermonde asked for a tour on the faces of a 8 x 8 x 8 cube. He gives it as a problem with a solution.

N. M. Gibbins. Chess in three and four dimensions. MG 28 (No. 279) (1944) 46 50. Gives knight's tour on 3 x 3 x 4 board -- an unpublished result due to E. Hubar Stockar of Geneva. This is the smallest 3 D board with a tour. Gives Maack's tour on 4 x 4 x 4 board.

Ian Stewart. Solid knight's tours. JRM 4:1 (Jan 1971) 1. Cites Dudeney. Gives a tour through the entire 8 x 8 x 8 cube by stacking 8 knight's paths.

T. W. Marlow. Closed knight tour of a 4 x 4 x 4 board. Chessics 29 & 30 (1987) 162. Inspired by Stewart.

5.F.4. OTHER CIRCUITS IN AND ON A CUBE
The number of Hamiltonian Circuits on the n-dimensional cube is the same as the number of Gray codes (see 7.M.3) and has been the subject of considerable research. I will not try to cover this in detail.
D. W. Crowe. The n dimensional cube and the Tower of Hanoi. AMM 63:1 (Jan 1956) 29 30.

E. N. Gilbert. Gray codes and paths on the n-cube. Bell System Technical Journal 37 (1958) 815-826. Shows there are 9 inequivalent circuits on the 4-cube and 1 on the n-cube for n = 1, 2, 3. The latter cases are sufficiently easy that they may have been known before this.

Allen F. Dreyer. US Patent 3,222,072 -- Block Puzzle. Filed: 11 Jun 1965; patented: 7 Dec 1965. 4pp + 2pp diagrams. 27 cubes on an elastic. The holes are straight or diagonal so that three consecutive cubes are either in a line or form a right angle. A solution is a Hamiltonian path through the 27 cells. Such puzzles were made in Germany and I was given one about 1980 (see Singmaster and Haubrich & Bordewijk below). Dreyer gives two forms.

Gardner. The binary Gray code. SA (Aug 1972) c= Knotted, chap. 2. Notes that the number of circuits on the n-cube, n > 4, is not known. SA (Apr 1973) reports that three (or four) groups had found the number of circuits on the 4-cube -- this material is included in the Addendum in Knotted, chap. 2, but none of the groups ever seem to have published their results elsewhere. Unfortunately, none of these found the number of inequivalent circuits since they failed to take all the equivalences into account -- e.g. for n = 1, 2, 3, 4, 5, their enumerations give: 2, 8, 96, 43008, 5 80189 28640 for the numbers of labelled circuits. Gardner's Addendum describes some further work including some statistical work which estimates the number on the 6-cube is about 2.4 x 10²⁵.

David Singmaster. A cubical path puzzle. Written in 1980 and submitted to JRM, but never published. For the 3 x 3 x 3 problem, the number, S, of straight through pieces (ignoring the ends) satisfies 2  S  11.

Mel A. Scott. Computer output, Jun 1986, 66pp. Determines there are 3599 circuits through the 3 x 3 x 3 cube such that the resulting string of 27 cubes can be made into a cube in just one way. But cf the next article which gives a different number??

Jacques Haubrich & Nanco Bordewijk. Cube chains. CFF 34 (Oct 1994) 12 15. Erratum, CFF 35 (Dec 1994) 29. Says Dreyer is the first known reference to the idea and that they were sold 'from about 1970' Reproduces the first page of diagrams from Dreyer's patent. Says his first version has a unique solution, but the second has 38 solutions. They have redone previous work and get new numbers. First, they consider all possible strings of 27 cubes with at most three in a line (i.e. with at most a single 'straight' piece between two 'bend' pieces and they find there are 98,515 of these. Only 11,487 of these can be folded into a 3 x 3 x 3 cube. Of these, 3654 can be folded up in only one way. The chain with the most solutions had 142 different solutions. They refer to Mel Scott's tables and indicate that the results correspond -- perhaps I miscounted Scott's solutions??
5.G. CONNECTION PROBLEMS
5.G.1. GAS, WATER AND ELECTRICITY
Dudeney. Problem 146 -- Water, gas, and electricity. Strand Mag. 46 (No. 271) (Jul 1913) 110 & (No. 272) (Aug 1913) 221 (c= AM, prob. 251, pp. 73 & 200 201). Earlier version is slightly more interesting, saying the problem 'that I have called "Water, Gas, and Electricity" ... is as old as the hills'. Gives trick solution with pipe under one house.

A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 -- BMC]. No. 96: The "three houses" problem, pp. 89-90 & 114. "Were all the houses connected up with all three supplies or not?" Answer is no -- one connection cannot be made.

Loyd, Jr. SLAHP. 1928. The three houses and three wells, pp. 6 & 87 88. "A puzzle ... which I first brought out in 1900 ..." The drawing is much less polished than Dudeney's. Trick solution with a pipe under one house, a bit differently laid out than Dudeney.

The Bile Beans Puzzle Book. 1933. No. 46: Water, gas & electric light. Trick solution almost identical to Dudeney.

Philip Franklin. The four color problem. In: Galois Lectures; Scripta Mathematica Library No. 5; Scripta Mathematica, Yeshiva College, NY, 1941, pp. 49-85. On p. 74, he refers to the graph as "the basis of a familiar puzzle, to join each of three houses with each of three wells (or in a modern version to a gas, water, and electricity plant)".

Leeming. 1946. Chap. 6, prob. 4: Water, gas and electricity, pp. 71 & 185. Dudeney's trick solution.

H. ApSimon. Note 2312: All modern conveniences. MG 36 (No. 318) (Dec 1952) 287 288. Given m houses and n utilities, the maximum number of non crossing connections is 2(m+n 2) and this occurs when all the resulting regions are 4 sided. He extends to p partite graphs in general and a special case.

John Paul Adams. We Dare You to Solve This! Op. cit. in 5.C. 1957? Prob. 50: Another enduring favorite appears below, pp. 30 & 49. Electricity, gas, water. Dudeney's trick solution.

Young World. c1960. P. 4: Crossed lines. Electricity, TV and public address lines. Trick solution with a line passing under a house.

T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751 753. Shows you can join 4 utilities to 4 houses on a torus without crossing.

5.H. COLOURED SQUARES AND CUBES, ETC.
5.H.1. INSTANT INSANITY = THE TANTALIZER

Note. Often the diagrams do not show all sides of the pieces so I cannot tell if one version is the same as another.

Frederick A. Schossow. US Patent 646,463 -- Puzzle. Applied: 19 May 1899; patented: 3 Apr 1900. 1p + 1p diagrams. Described in S&B, p. 38, which also says it is described in O'Beirne, but I don't find it there?? Four cubes with suit patterns. The net of each cube is shown. The fourth cube has three clubs.

George Duncan Moffat. UK Patent 9810 -- Improvements in or relating to Puzzle-apparatus. Applied: 28 May 1900; accepted: 30 Jun 1900. 2pp + 1p diagrams. For a six cube version with "letters R, K, B, W, F and B-P, the initials of the names of General Officers of the South African Field Force."

Joseph Meek. UK Patent 2775 -- Improved Puzzle Game. Applied: 5 Feb 1909; complete specification: 16 Jun 1909; accepted: 3 Feb 1910. 2pp + 1p diagrams. A four cube version with suit patterns. His discussion seems to describe the pieces drawn by Schossow.

Slocum. Compendium. Shows: The Great Four Ace Puzzle (Gamage's, 1913); Allies Flag Puzzle (Gamage's, c1915); Katzenjammer Puzzle (Johnson Smith, 1919).

Edwin F. Silkman. US Patent 2,024,541 -- Puzzle. Applied: 9 Sep 1932; patented: 17 Dec 1935. 2pp + 1 p diagrams. Four cubes marked with suits. The net of each cube is shown. The third cube has three hearts. This is just a relabelling of Schossow's pattern, though two cubes have to be reflected which makes no difference to the solution process.

E. M. Wyatt. The bewitching cubes. Puzzles in Wood. (Bruce Publishing, Co., Milwaukee, 1928) = Woodcraft Supply Corp., Woburn, Mass., 1980, p. 13. A six cube, six way version.

Abraham. 1933. Prob. 303 -- The four cubes, p. 141 (100). 4 cube version "sold ... in 1932".

A. S. Filipiak. Four ace cube puzzle. 100 Puzzles, How do Make and How to Solve Them. A. S. Barnes, NY, (1942) = Mathematical Puzzles, and Other Brain Twisters; A. S. Barnes, NY, 1966; Bell, NY, 1978; p. 108.

Leeming. 1946. Chap. 10, prob. 9: The six cube puzzle, pp. 128 129 & 212. Identical to Wyatt.

F. de Carteblanche [pseud. of Cedric A. B. Smith]. The coloured cubes problem. Eureka 9 (1947) 9 11. General graphical solution method, now the standard method.

T. H. O'Beirne. Note 2736: Coloured cubes: A new "Tantalizer". MG 41 (No. 338) (Dec 1957) 292-293. Cites Carteblanche, but says the current version is different. Gives a nicer version.

T. H. O'Beirne. Note 2787: Coloured cubes: a correction to Note 2736. MG 42 (No. 342) (Dec 1958) 284. Finds more solutions than he had previously stated.

Norman T. Gridgeman. The 23 colored cubes. MM 44:5 (Nov 1971) 243-252. The 23 colored cubes are the equivalence classes of ways of coloring the faces with 1 to 6 colors. He cites and describes some later methods for attacking Instant Insanity problems.

Jozsef Bognár. UK Patent Application 2,076,663 A -- Spatial Logical Puzzle. Filed 28 May 1981; published 9 Dec 1981. Cover page + 8pp + 3pp diagrams. Not clear if the patent was ever granted. Describes Bognár's Planets, which is a four piece instant insanity where the pieces are spherical and held in a plastic tube. This was called Bolygok in Hungarian and there is a reference to an earlier Hungarian patent. Also describes his version with eight pieces held at the corners of a plastic cube.

5.H.2. MACMAHON PIECES
Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include MacMahon puzzles as one class.

I have just added the Carroll result that there are 30 six-coloured cubes, but this must be older??

Frank H. Richards. US Patent 331,652 -- Domino. Applied: 13 Jun 1885; patented: 1 Dec 1885. 2pp + 2pp diagrams. Cited by Gardner in Magic Show, but with date 1895. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. For triangular matching games, specifically showing the MacMahon 5-coloured triangles, but considering reflections as equivalences, so he has 35 pieces. [One of the colours is blank and hence Gardner said it was a 4-colouring.]

Carroll-Wakeling. c1890? Prob. 15: Painting cubes, pp. 18-19 & 67. This is one of the problems on undated sheets of paper that Carroll sent to Bartholomew Price. How many ways can one six-colour a cube? Wakeling gives a solution, but this apparently is not on Carroll's MS.

Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 3927 A.D. 1892 -- Appliances to be used in Playing a New Class of Games. Applied: 29 Feb 1892; Complete Specification Left: 28 Nov 1892; Accepted: 28 Jan 1893. 5pp + 2pp diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. Describes the 24 triangles with four types of edge and mentions other numbers of edge types. Describes various games and puzzles.

Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 8275 A.D. 1892 -- Appliances for New Games of Puzzles. Applied: 2 May 1892; Complete Specification Left: 31 Jan 1893; Accepted: 4 Mar 1893. 2pp. 27 cubes with three colours, opposite faces having the same colour. Similar sets of 8, 27, 64, etc. cubes. Various matching games suggested. Using six colours and all six on each cube gives 30 cubes -- the MacMahon Cubes. Gives a complex matching problem of making two 2 x 2 x 2 cubes. Paul Garcia (email of 15 Nov 2002) commented: "8275 describes 2 different sets of blocks, using either three colours or six colours. The three colour blocks form a set of 27 that can be assembled into a large cube with single coloured faces and internal contact faces matching. For the six colour cubes, the puzzle suggested is to pick out two associated cubes, and find the sixteen cubes that can be assembled to make a copy of each. Not quite Mayblox, although using the same colouring system."

James Dalgety. R. Journet & Company A Brief History of the Company & its Puzzles. Published by the author, North Barrow, Somerset, 1989. On p. 13, he says Mayblox was patented in 1892. In an email on 12 Nov 2002, he cited UK Patent 8275.

Anon. Report: "Mathematical Society, February 9". Nature 47 (No. 1217) (23 Feb 1893) 406. Report of MacMahon's talk: The group of thirty cubes composed by six differently coloured squares.

See: Au Bon Marché, 1907, in 5.P.2, for a puzzle of hexagons with matching edges.

Manson. 1911. Likoh, pp. 171-172. MacMahon's 24 four-coloured isosceles right triangles, attributed to MacMahon.

"Toymaker". The Cubes of Mahomet Puzzle. Work, No. 1447 (9 Dec 1916) 168. 8 six-coloured cubes to be assembled into a cube with singly-coloured faces and internal faces to have matching colours.

P. A. MacMahon. New Mathematical Pastimes. CUP, 1921. The whole book deals with variations of the problem and calculates the numbers of pieces of various types. In particular, he describes the 24 4-coloured triangles, the 24 3-coloured squares, the MacMahon cubes, some right-triangular and hexagonal sets and various subsets of these. With n colours, there are n(n²+2)/3 triangles, n(n+1)(n² n+2)/4 squares and n(n+1)(n⁴-n³+n²+2)/6 hexagons. [If one allows reflectional equivalence, one gets n(n+1)(n+2)/6 triangles, n(n+1)(n²+n+2)/8 squares and n(n+1)(n⁴-n³+4n²+2)/12 hexagons. Problem -- is there an easy proof that the number of triangles is BC(n+2, 3)?] On p. 44, he says that Col. Julian R. Jocelyn told him some years ago that one could duplicate any cube with 8 other cubes such that the internal faces matched.

Slocum. Compendium. Shows Mayblox made by R. Journet from Will Goldston's 1928 catalogue.

F. Winter. Das Spiel der 30 bunten Würfel MacMahon's Problem. Teubner, Leipzig, 1934, 128pp. ??NYR.

Clifford Montrose. Games to play by Yourself. Universal Publications, London, nd [1930s?]. The coloured squares, pp. 78-80. Makes 16 squares with four-coloured edges, using five colours, but there is no pattern to the choice. Uses them to make a 4 x 4 array with matching edges, but seems to require the orientations to be fixed.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 6. There are twelve ways to colour the edges of a pentagon, when rotations and reflections are considered as equivalences. Can you colour the edges of a dodecahedron so each of these pentagonal colourings occurs once? [If one uses tiles, one has to have reversible tiles.] Solution says there are three distinct solutions and describes them by describing contacts between 10 pentagons forming a ring around the equator.

Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.] 7 (Oct & Dec 1960) 97-106 & 144-153. Pp. 101-104 discusses MacMahon triangles, squares and hexagons.

T. H. O'Beirne. Puzzles and paradoxes 5: MacMahon's three-colour set of squares. New Scientist 9 (No. 220) (2 Feb 1961) 288-289. Finds 18 of the 20 possible monochrome border patterns.

Gardner. SA (Mar 1961) = New MD, Chap. 16. MacMahon's 3-coloured squares and his cubes. Addendum in New MD cites Feldman, below.

Gary Feldman. Documentation of the MacMahon Squares Problem. Stanford Artificial Intelligence Project Memo No. 12, Stanford Computation Center, 16 Jan 1964. ??NYS Finds 12,261 solutions for the 6 x 4 rectangle with monochrome border -- but see Philpott, 1982, for 13,328 solutions!!

Gardner. SA (Oct 1968) = Magic Show, Chap. 16. MacMahon's four-coloured triangles and numerous variants.

Wade E. Philpott. MacMahon's three-color squares. JRM 2:2 (1969) 67-78. Surveys the topic and repeats Feldman's result.

N. T. Gridgeman, loc. cit. in 5.H.1, 1971, covers some ideas on the MacMahon cubes.

J. J. M. Verbakel. Digitale tegels (Digital tiles). Niet piekeren maar puzzelen (name of a puzzle column). Trouw (a Dutch newspaper) (1 Feb 1975). ??NYS -- described by Jacques Haubrich; Pantactic patterns and puzzles; CFF 34 (Oct 1994) 19-21. There are 16 ways to 2 colour the edges of a square if one does not allow them to rotate. Assemble these into a 4 x 4 square with matching edges. There are 2,765,440 solutions in 172,840 classes of 16. One can add further constraints to yield fewer solutions -- e.g. assume the 4 x 4 square is on a torus and make all internal lines have a single colour.

Gardner. Puzzling over a problem solving matrix, cubes of many colours and three dimensional dominoes. SA 239:3 (Sep 1978) 20 30 & 242 c= Fractal, chap. 11. Good review of MacMahon (photo) and his coloured cubes. Bibliography cites recent work on Mayblox, etc.

Wade E. Philpott. Instructions for Multimatch. Kadon Enterprises, Pasadena, Maryland, 1982. Multimatch is just the 24 MacMahon 3-coloured squares. This surveys the history, citing several articles ??NYS, up to the determination of the 13,328 solutions for the 6 x 4 rectangle with monochrome border, by Hilario Fernández Long (1977) and John W. Harris (1978).

Torsten Sillke. Three 3 x 3 matching puzzles. CFF 34 (Oct 1994) 22-23. He has wanted an interesting 9 element subset of the MacMahon pieces and finds that of the 24 MacMahon 3-coloured squares, just 9 of them contain all three colours. He considers both the corner and the edge versions. The editor notes that a 3 x 3 puzzle has 36 x 32/2 = 576 possible edge contacts and that the number of these which match is a measure of the difficulty of the puzzle, with most 3 x 3 puzzles having 60 to 80 matches. The corner version of Sillke's puzzle has 78 matches and one solution. The edge version has 189 matches and many solutions, hence Sillke proposes various further conditions.