Sources page biographical material

6.W. BURR PUZZLES
When assembled, a burr looks like three sticks crossing orthogonally, forming a 'star' with six points at the vertices of an octahedron. Slocum says Wyatt [Puzzles in Wood, 1928, op. cit. in 5.H.1] is the first to use the word 'burr'. Collins, Book of Puzzles, 1927, p. 135, calls them "Cluster, Parisian or Gordian Knot Puzzles" and states: "it is believed that they were first made in Paris, if, indeed, they were not invented there." Since about 1990, there has been considerable development in new types of burr which use plates or boards rather than sticks, or whose central volume is subdivided more (cf in 6.W.1).

See S&B, pp. 62 85.

See also 6.BJ.
6.W.1. THREE PIECE BURR
Most of these have three pieces which are rectangular in cross-section (1 x 3 x 5) with slots of the same size and some of the pieces have notches from the slot to the outside. When one piece is pushed, it slides, revealing its notch. When placed properly, this allows a second piece to slide off and out.

In the 1990s, a more elaborate type of three piece burr appeared. These have three 3 x 3 x 5 pieces which intersect in a central 3 x 3 x 3 region. Within this region, some of the unit cubes are not present, which allows sliding of the pieces. Some versions of the puzzle permit twisting of pieces though this usually requires a bit of rounding of edges and the actual examples tend to break, so these are not as acceptable.

Crambrook. 1843. P. 5, no. 4: Puzzling Cross 3 pieces. This seems likely to be a three piece burr, but perhaps is in 6.W.3 -- ?? It is followed by "Maltese Cross 6 pieces".

Edward Hordern's collection has examples in ivory from 1850-1900.

Hoffmann. 1893. Chap. III, no. 35: The cross keys or three piece puzzle, pp. 106 & 139 = Hoffmann-Hordern, pp. 104-105, with photo. One piece has an extra small notch which does not appear in other versions where the dimensions are better chosen. I have recently acquired an example which appears identical to the illustrations but does not have the extra notch - this came from a Jaques puzzle box, c1900, and Dalgety has several examples of such boxes with the solution, where the puzzle is named The Cross Keys Puzzle (cf discussion at the beginning of Section 11). The photo on p. 105 is an assembled version, with verbal instructions, by Jaques & Son, 1880-1895 (but Jaques was producing them up to at least c1910). Hordern Collection, p. 67, shows Le Noeud Mystérieux, 1880 1905, with a pictorial solution and this does not have the extra notch.

Benson. 1904. The cross keys puzzle, pp. 205 206.

Pearson. 1907. Part III, no. 56: The cross keys, pp. 56 & 127 128.

Anon. A puzzle in wood. Hobbies 31 (No. 795) (7 Jan 1911) 345. Three piece burr with small extra notch as in Hoffmann.

Anon. Woodwork Joints. Evans, London, (1918), 2nd ed., 1919. [I have also seen a 4th ed., 1925, which is identical to the 2nd ed., except for advertising pages at the end.] A mortising puzzle, pp. 197 199.

Collins. Book of Puzzles. 1927. Pp. 136-137: The cross keys puzzle.

E. M. Wyatt. Three piece cross. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 24 25.

Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared in 1908, but versions continued until the 1950s. This looks like 1930s?? 3-Piece Mortise with thin pieces.

A. S. Filipiak. Burr puzzle. Mathematical Puzzles, 1942, op. cit. in 5.H.1, p. 101.

Dic Sonneveld seems to be the first to begin designing three piece burrs of the more elaborate style, perhaps about 1985. Trevor Wood has made several examples for sale.

Bill Cutler. Email announcement to NOBNET on 27 Jan 1999. He has begun analysing the newer style of three piece burr, excluding twist moves. His first stage has examined cases where the centre cube of the central region is occupied and the piece this central cube belongs to has no symmetry. He finds 202 x 10⁹ assemblies (I'm not sure if this is an exact figure) and there are 33 level-8 examples (i.e. where it takes 8 moves to remove the first piece); 6674 level-7 examples; 73362 level-6 examples. He thinks this is about 70% of the total and it is already about six times the number of cases considered for the six piece burr (see 6.W.2).

Bill Cutler. Christmas letter of 4 Dec 1999. Says he has completed the above analysis and found 25 x 10¹⁰ possibilities, which took 225 days on a workstation. The most elaborate examples require 8 moves to get a piece out and there are 80 of these. He used one for his IPP19 puzzle. He has a website with many of his results on burrs, etc.: www.billcutlerpuzzles.com .

6.W.2. SIX PIECE BURR = CHINESE CROSS
The usual form of these has six sticks, 2 x 2 x 6 (or 8), which have various notches in them. In the 1990s, new forms were introduced, using plates or boards. One version makes an open frame shape, something like a 3 x 3 x 3 chessboard. In the other, 1 x 4 x 6 boards are paired side by side and the result looks like a classic six-piece burr with the end rectangle divided lengthwise rather than crosswise. See also 6.W.7.
Jurgis Baltrušaitis. Anamorphoses ou magie artificielle des effets merveilleux. Olivier Perrin Éditeur, Paris, 1969. On pp. 110-116 & 184 is a discussion of a 1698 engraving "L'Académie des Sciences et des Beaux Arts" by Sébastien Leclerc (or Le Clerc). In the right foreground is an object looking like a six piece burr. James Dalgety discusses this in his Latest news on oldest puzzles; Lecture to Second Meeting on the History of Recreational Mathematics, 1 Jun 1996. This image also exists in a large painted version (950 x 480 mm) which is more precise and more legible in many details, so it is supposed that the engraving was done in conjunction with the painting. Though it was normal for a notable painting to be turned into an engraving, the opposite sometimes happened and Leclerc was a famous engraver. The painter is unknown. The divisions between the pairs of pieces of the 'burr' are pretty clear in the engraving, but two of them are not visible in the painting. The 'burr' is also not quite correctly drawn, but all in all, it seems pretty convincing. James Dalgety was the first to discover this picture and he has a copy of the engraving, but has not been able to locate the painting, though it was in the Bernard Monnier Collection exhibited at the Musée des Arts Decoratifs in Paris in 1975/76.

Camille Frémontier. Sébastien Leclerc and the British Encyclopeaedists. Sphæra [Newsletter of the Museum of the History of Science, Oxford] 6 (Aut 1997) 6-7. Discusses the Leclerc engraving which was used as the frontispiece to several encyclopedias, the earliest being Chambers Cyclopaedia of 1728.

Minguet. 1733. Pp. 103-105 (1755: 51-52; 1822: 122-124; 1864: 103-104). Pieces diagrammed. One plain key piece.

Catel. Kunst-Cabinet. 1790. Die kleine Teufelsklaue, p. 10 & fig. 16 on plate I. Figure shows it assembled and fails to draw one of the divisions between pieces. Description says it is 6 pieces, 2 inches long, from plum wood and costs 3 groschen (worth about an English penny of the time). (See also pp. 9-10, fig. 20 on plate I for Die grosse Teufelsklaue -- the 'squirrelcage'.)

Bestelmeier. 1801. Item 147: Die kleine Teufelsklaue. (Note -- there is another item 147 on the next plate.) Only shows it assembled. Brief text may be copying part of Catel. See also the picture for item 1099 which looks like a six piece burr included in a set of puzzles. (See also Item 142: Die grosse Teufelsklaue.)

Edward Hordern's collection has examples, called The Oak of Old England, from c1840.

Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these might be here or in 6.W.4 or 6.W.5 -- ??

Magician's Own Book. 1857. Prob. 1: The Chinese cross, pp. 266-267 & 291. One plain key piece. Not the same as in Minguét.

Landells. Boy's Own Toy-Maker. 1858. Pp. 137-139. Identical to Magician's Own Book.

Book of 500 Puzzles. 1859. 1: The Chinese cross, pp. 80-81 & 105. Identical to Magician's Own Book.

A. F. Bogesen (1792 1876). In the Danish Technical Museum, Helsingør (= Elsinore) are a number of wooden puzzles made by him, including a 6 piece burr, a 12 piece burr, an Imperial Scale? and a complex (trick??) joint.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 23: The Chinese Cross, pp. 399 & 439. Identical to Magician's Own Book, except one diagram in the solution omits two labels.

Boy's Own Conjuring Book. 1860. Prob. 1: The Chinese cross, pp. 228 & 254. Identical to Magician's Own Book.

Hoffmann. 1893. Chap. III, no. 36: The nut (or six piece) puzzle, pp. 106 & 139 140 = Hoffmann-Hordern, pp. 104-106. Different pieces than in Minguét and Magician's Own Book.

Dudeney. Prob. 473 -- Chinese cross. Weekly Dispatch (23 Nov & 7 Dec 1902), both p. 13. "There is considerable variety in the manner of cutting out the pieces, and though the puzzle has been given in some of the old books, I have purposely presented it in a form that has not, I believe, been published."

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Chinese Cross," a puzzle of undoubted Oriental origin that was formerly brought from China by travellers as a curiosity, but for a long time has had a steady sale in this country."

Wehman. New Book of 200 Puzzles. 1908. The Chinese cross, pp. 40-41. = Magician's Own Book.

Dudeney. The world's best puzzles. 1908. Op. cit. in 2. P. 779 shows a '"Chinese Cross" which ... is of great antiquity.'

Oscar W. Brown. US Patent 1,225,760 -- Puzzle. Applied: 27 Jun 1916; patented: 15 May 1917. 3pp + 1p diagrams. Coffin says this is the earliest US patent, with several others following soon after.

Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. Eastern joint puzzle, pp. 196 197: Two versions using different pieces. Six piece joint puzzle, pp. 199 200. Another version.

Western Puzzle Works, 1926 Catalogue. No. 86: 6 piece Wood Block. Several other possible versions -- see 6.W.7.

E. M. Wyatt. Six piece burr. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 27 28. Describes 17 versions from 13 types of piece.

A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in 5.H.1, pp. 79 87. 73 versions from 38 types of piece.

William H. [Bill] Cutler. The six piece burr. JRM 10 (1977 78) 241 250. Complete, computer assisted, analysis, with help from T. H. O'Beirne and A. C. Cross. Pieces are considered as 'notchable' if they can be made by a sequence of notches, which are produced by two saw cuts and then chiselling out the space between them. Otherwise viewed, notches are what could be produced by a wide cutter or router. There are 25 of these which can occur in solutions. (In 1994, he states that there are a total of 59 notchable pieces and diagrams all of them.) One can also have more general pieces with 'right-angle notches' which would require four chisel cuts -- e.g. to cut a single 1 x 1 x 1 piece out of a 2 x 2 x 8 rod. Alternatively, one can glue cubes into notches. There are 369 which can occur in solutions. (In 1994, he states that there are 837 pieces which produce 2225 different oriented pieces, and he lists them all.) He only considers solid solutions -- i.e. ones where there are no internal holes. He finds and lists the 314 'notchable' solutions. There are 119,979 general solutions.

C. Arthur Cross. The Chinese Cross. Pentangle, Over Wallop, Hants., UK, 1979. Brief description of the solutions in the general case, as found by Cutler and Cross.

S&B, p. 83, describes holey burrs.

W. H. [Bill] Cutler. Christmas letter, 1987. Sketches results of his (and other's) search for holey burrs with notchable pieces.

Bill Cutler. Holey 6 Piece Burr! Published by the author, Palatine, Illinois. (1986); with addendum, 1988, 48pp. He is now permitting internal holes. Describes holey burrs with notchable pieces, particularly those with multiple moves to release the first piece.

Bill Cutler. A Computer Analysis of All 6-Piece Burrs. Published by the author, ibid., 1994. 86pp. Sketches complete history of the project. (I have included a few details in the description of his 1977/78 article, above.) In 1987, he computed all the notchable holey solutions, using about 2 months of PC AT time, finding 13,354,991 assemblies giving 7.4 million solutions. Two of these were level 10 -- i.e. they require 10 moves to remove the first piece (or pieces), but the highest level occurring for a unique solution was 5. After that he started on the general holey burrs and estimated it would take 400 years of PC AT time -- running at 8 MHz. After some development, the actual time used was about 62.5 PC AT years, but a lot of this was done on by Harry L. Nelson during idle time on the Crays at Lawrence Livermore Laboratories, and faster PCs became available, so the whole project only took about 2½ years, being completed in Aug 1990 and finding 35,657,131,235 assemblies. He hasn't checked if all assemblies come apart fully, but he estimates there are 5.75 billion solutions. He estimates the project used 45 times the computing power used in the proof of the Four Color Theorem and that the project would only take two weeks on the eight RS6000 workstations he now supervises. Some 70,000 high-level solutions were specifically saved and can be obtained on disc from him. The highest level found was 12 and the highest level for a unique solution was 10. See 6.W.1 for a continuation of this work. He has a website with many of his results on burrs, etc.: www.billcutlerpuzzles.com .

Bill Cutler & Frans de Vreugd. Information leaflet accompanying their separate IPP22 puzzles, 2002. In 2001, they did an analysis of six-board burrs, of the type where the boards are paired side by side. There are 4096 possible such boards, but only 219 usable boards occur. They looked at all combinations of six of these and found 14,563,061,989 assemblies. Of these, the highest level found was 13.

6.W.3. THREE PIECE BURR WITH IDENTICAL PIECES
See S&B, p. 66.
Crambrook. 1843. P. 5, no. 4: Puzzling Cross 3 pieces. This seems likely to be a three piece burr, but perhaps is in 6.W.1 -- ?? It is followed by "Maltese Cross 6 pieces".

Wilhelm Segerblom. Trick wood joining. SA (1 Apr 1899) 196.

6.W.4. DIAGONAL SIX PIECE BURR = TRICK STAR
This version often looks like a stellated rhombic dodecahedron. It has two basic forms, one with a key piece; the other with all pieces identical, which assembles as two groups of three.

See S&B, p. 78.

Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these belong here or in 6.W.2 or 6.W.5 -- ??

The Youth's Companion. 1875. [Mail order catalogue.] Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois, 1977?, p. 19. Star Puzzle. The picture does not show which form it is. Slocum's Compendium also shows this.

Samuel P. Chandler. US Patent 393,816 -- Puzzle. Applied: 9 Mar 1888; patented: 23 Apr 1888. 1p + 1p diagrams. Coffin says this is the earliest version, but it is more complex than usual, with 12 pieces, and has a key piece.

John S. Pinnell. US Patent 774,197 -- Puzzle. Applied: 9 Oct 1902; patented: 8 Nov 1904. 2pp + 2pp diagrams. Coffin notes that this extends the idea to 102 pieces!

William E. Hoy. US Patent 766,444 -- Puzzle Ball. Applied: 16 Oct 1902; patented: 2 Aug 1904. 2pp + 2pp diagrams. Spherical version with a key piece.

George R. Ford. US Patent 779,121 -- Puzzle. Applied: 16 May 1904; patented: 3 Jan 1905. 1p + 1p diagrams. With square rods, all identical. He shows assembly by inserting a last piece rather than joining two groups of three.

Anon. Simple wood puzzle. Hobbies 31 (No. 786) (5 Nov 1910) 127. With key piece.

E. M. Wyatt. Woodwork puzzles. Industrial Arts Magazine 12 (1923) 326 327. Version with a key piece and square rods.

Collins. Book of Puzzles. 1927. The bonbon or nut puzzle, pp. 137-139.

Iffland Frères (Lausanne). Swiss Patent 245,402 -- Zusammensetzspiel. Received: 19 Nov 1945; granted: 15 Nov 1946; published: 1 Jul 1947. 2pp + 1p diagrams. Stellated rhombic dodecahedral version with a key piece. (Coffin says this is the first to use this shape, although Slocum has a version c1875.)

6.W.5. SIX PIECE BURR WITH IDENTICAL PIECES
One form has six identical pieces and all move outward or inward together. Another form with flat notched pieces has one piece with an extra notch or an extended notch which allows it to fit in last, either by sliding or twisting, but this is not initially obvious. This form is sometimes made with equal pieces so that it can only be assembled by force, perhaps after steaming, and it then makes an unopenable money box. This might be considered under 11.M.
Edward Hordern's collection has a version with one piece a little smaller than the rest from c1800.

Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these belong here or in 6.W.2 or 6.W.4 -- ??

C. Baudenbecher catalogue, c1850s. Op. cit. in 6.W.7. This has an example of the six equal flat pieces making an unopenable(?) money box.

F. Chasemore. Some mechanical puzzles. In: Hutchison; op. cit. in 5.A; 1891, chap. 70, part 1, pp. 571 572. Item 5: The puzzle box, p. 572. Six U pieces make a uniformly expanding cubical box.

Hoffmann. 1893. Chap. III, no.33: The bonbon nut puzzle, pp. 104 & 138 = Hoffmann Hordern, pp. 102-103, with photo. One piece has an extra notch to simplify the assembly. Photo on p. 103 shows an example, almost certainly by Jaques & Son, 1860-1895.

Burnett Fallow. How to make a puzzle money-box. The Boy's Own Paper 15 (No. 755) (1 Jul 1893) 638. Equal flat notched pieces forced together to make an unopenable box.

Burnett Fallow. How to make a puzzle picture-frame. The Boy's Own Paper 16 (No. 815) (25 Aug 1894) 749. Each corner has the same basic forced construction as used in the puzzle money-box.

Benson. 1904. The bonbon nut puzzle, p. 204.

Bartl. c1920. Several versions on p. 306.

Western Puzzle Works, 1926 Catalogue. Last page shows 20 Chinese Wood Block Puzzles, High Grade. Some of these are of the present type.

Collins. Book of Puzzles. 1927. The bonbon or nut puzzle, pp. 137-139. As in Hoffmann.

Iona & Robert Opie and Brian Alderson. Treasures of Childhood. Pavilion (Michael Joseph), London, 1989. P. 158 shows a "cluster puzzle which Professor Hoffman [sic] names the 'Nut (or Six piece) Puzzle', but which is usually called 'The Maltese Puzzle'."

6.W.6. ALTEKRUSE PUZZLE
William Altekruse. US Patent 430,502 -- Block-Puzzle. Applied: 3 Apr 1890; patented: 17 Jun 1890. 1p + 1p diagrams. Described in S&B, p. 72. The standard version has 12 pieces, but variations discovered by Coffin have 14, 36 & 38 pieces.

Western Puzzle Works, 1926 Catalogue. No. 112: 12 piece Wood Block. Possibly Altekruse.

6.W.7. OTHER BURRS
See also 6.BJ for other 3D dissections. I have avoided repeating items, so 6.BJ should also be consulted if you are reading this section.
Catel. Kunst-Cabinet. 1790. Die grosse Teufelsklaue, pp. 9-10 & fig. 20 on plate I. 24 piece 'squirrel cage'. Cost 16 groschen.

Bestelmeier. 1801. Item 142: Die grosse Teufelsklaue. The 'squirrelcage', identical to Catel, with same drawing, but reversed. Text may be copying some of Catel.

C. Baudenbecher, toy manufacturer in Nuremberg. Sample book or catalogue from c1850s. Baudenbecher was taken over by J. W. Spear & Sons in 1919 and the catalogue is now in the Spear's Game Archive, Ware, Hertfordshire. It comprises folio and double folio sheets with finely painted illustrations of the firm's products. One whole folio page shows about 20 types of wooden interlocking puzzles, including most of the types mentioned elsewhere in this section and in 6.W.5 and 6.BJ. Until I get a picture, I can't be more specific.

The Youth's Companion. 1875. [Mail order catalogue.] Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois, 1977?, p. 19. Shows a 'woodchuck' type puzzle, called White Wood Block Puzzle, from The Youth's Companion, 1875. I can't see how many pieces it has: 12 or 18?? Slocum's Compendium also shows this.

Slocum. Compendium. Shows: "Mystery", Magic "Champion Puzzle" and "Puzzle of Puzzles" from Bland's Catalogue, c1890.

The first looks like a 6 piece burr with circular segments added to make it look like a ball. So it may be a 6 piece burr in disguise. See also Hoffmann, Chap. III, no. 38, pp. 107 108 & 141 142 = Hoffmann-Hordern, pp. 106-108 = Benson, p. 205.

The second is a six piece puzzle, but the pieces are flattish and it may be of the type described in 6.W.5.

The third is complex, with perhaps 18 pieces.

Bartl. c1920. Several versions on pp. 306-307, including some that are in 6.W.5 and some 'Chinese block puzzles'.

Western Puzzle Works, 1926 Catalogue. Shows a number of burrs and similar puzzles.

No. 86: 6 piece Wood Block.

No. 112: 12 piece Wood Block. Possibly Altekruse.

No. 212: 11 piece Wood Block

The last page shows 20 Chinese Wood Block Puzzles, High Grade. Some of these are burrs.

Collins. Book of Puzzles. 1927. Other cluster puzzles, pp. 139-142. Describes and illustrates: The cluster; The cluster of clusters; The gun cluster; The point cluster; The flat cluster; The cluster (or secret) table; The barrel; The Ball; The football. All of these have a key piece.

Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escher-puzzle; Pythagoras (Amsterdam) (1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -- ??) (1992) 12-13. Two Penrose tribars made into an impossible 5-piece burr.

6.X. ROTATING RINGS OF POLYHEDRA
Generally, these have edge to edge joints. 'Jacob's ladder' joints are used by Engel -- see 11.L for other forms of this joint.
I am told these may appear in Fedorov (??NYS).

Max Brückner. Vielecke und Vielfläche. Teubner, Leipzig, 1900. Section 162, pp. 215 216 and Tafel VIII, fig. 4. Describes rings of 2n tetrahedra joined edge to edge, called stephanoids of the second order. The figure shows the case n = 5.

Paul Schatz. UK Patent 406,680 -- Improvements in or relating to Boxes or Containers. Convention date (Germany): 10 Dec 1931; application date (in UK): 19 Jul 1932; accepted: 19 Feb 1934. 6pp + 6pp diagrams. Six and four piece rings of prisms which fold into a box.

Paul Schatz. UK Patent 411,125 -- Improvements in Linkwork comprising Jointed Rods or the like. Convention Date (Germany): 31 Aug 1931; application Date (in UK): 31 Aug 1932; accepted: 31 May 1934. 3p + 6pp diagrams. Rotating rings of six tetrahedra and linkwork versions of the same idea, similar to Flowerday's Hexyflex.

Ralph M. Stalker. US Patent 1,997,022 -- Advertising Medium or Toy. Applied: 27 Apr 1933; patented: 9 Apr 1935. 3pp + 2pp diagrams. "... a plurality of tetrahedron members or bodies flexibly connected together." Shows six tetrahedra in a ring and an unfolded pattern for such objects. Shows a linear form with 14 tetrahedra of decreasing sizes.

Sidney Melmore. A single sided doubly collapsible tessellation. MG 31 (No. 294) (1947) 106. Forms a Möbius strip of three triangles and three rhombi, which is basically a flexagon (cf 6.D). He sees it has two distinct forms, but doesn't see the flexing property!! He describes how to extend these hexagons into a tessellation which has some resemblance to other items in this section.

Alexander M. Shemet. US Patent 2,688,820 -- Changeable Display Amusement Device. Applied: 25 Jul 1950; patented: 14 Sep 1954. 2pp + 2pp diagrams. Basically a rotating ring of six tetrahedra, but says 'at least six'. Gives an unfolded version or net for making it and a mechanism for flexing it continually. Cites Stalker.

Wallace G. Walker invented his "IsoAxis" ® in 1958 while a student at Cranbrook Academy of Art, Michigan. This is approximately a ring of ten tetrahedra. He obtained a US Patent for it in 1967 -- see below. In 1973(?) he sent an example to Doris Schattschneider who soon realised that the basic idea was a ring of tetrahedra and that Escher tessellations could be adapted to it. They developed the idea into "M. C. Escher Kaleidocycles", published by Ballantine in 1977 and reprinted several times since.

Douglas Engel. Flexahedrons. RMM 11 (Oct 1962) 3 5. These have 'Jacob's ladder' hinges, not edge to edge hinges. He says he invented these in Fall, 1961. He formed rings of 4, 6, 7, 8 tetrahedra and used a diagonal joining to make rings of 4 and 6 cubes.

Wallace G. Walker. US Patent 3,302,321 -- Foldable Structure. Filed: 16 Aug 1963; issued: 7 Feb 1967. 2pp + 6pp diagrams.

Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Solid Flexagons, pp. 81 84. Based on Engel, but only gives the ring of 6 tetrahedra.

D. Engel. Flexing rings of regular tetrahedra. Pentagon 26 (Spring 1967) 106 108. ??NYS -- cited in Schaaf II 89 -- write Engel.

Paul Bethell. More Mathematical Puzzles. Encyclopædia Britannica International, London, 1967. The magic ring, pp. 12-13. Gives diagram for a ten-tetrahedra ring, all tetrahedra being regular.

Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. Presents versions of the flexing cubes and the 'Shinsei Mystery'. [Jan de Geus has sent a photocopy of some of this but it does not cover this topic.]

Jan Slothouber. Flexicubes -- reversible cubic shapes. JRM 6 (1973) 39 46. As above.

Frederick George Flowerday. US Patent 3,916,559 -- Vortex Linkages. Filed: 12 Aug 1974 (23 Aug 1973 in UK); issued: 4 Nov 1975. Abstract + 2pp + 3pp diagrams. Mostly shows his Hexyflex, essentially a six piece ring of tetrahedra, but with just four edges of each tetrahedron present. He also shows his Octyflex which has eight pieces. Text refers to any even number  6.

Naoki Yoshimoto. Two stars in a cube (= Shinsei Mystery). Described in Japanese in: Itsuo Sakane; A Museum of Fun; Asahi Shimbun, Tokyo, 1977, pp. 208 210. Shown and pictured as Exhibit V 1 with date 1972 in: The Expanding Visual World -- A Museum of Fun; Exhibition Catalogue, Asahi Shimbun, Tokyo, 1979, pp. 102 & 170 171. (In Japanese). ??get translated??

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 63-66. Describes Walkers IsoAxis and rotating rings of six and eight tetrahedra.