Sources page biographical material

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.