The goal is to arrange the pieces in some simple configuration such that their features respect a given rule (or rules) - form a pattern - occasionally but not typically enforced by mechanical means. For example, "piece features at specific points (e.g. edges or corners) match (or complement, or dismatch) those at corresponding points on abutting pieces" or "the heights (or numeric values) of aligned pieces total a specific constant" or "defined sets of faces have all distinct (or equal) features." Sometimes, as in the case of matchstick puzzles or the "Eight Queens" puzzle, the individual pieces are indistinguishable, but their arrangement in a particular pattern is paramount.

Unlike with Assembly puzzles, laying the pieces out or into a tray, or stacking them together to form a particular shape, is not the primary focus, though it might contribute to the difficulty - it is the pattern rule which provides the chief challenge. Success is usually determined by careful inspection, checking to see that the given rule has been everywhere satisfied.

You might see some puzzles in this section that remind you of jigsaw puzzles - but the key difference here is that the edge features are compatible with several other potential mates and there are usually several other pieces that match any given piece. The individual pieces can be easily arranged in many different ways that partially satisfy the rule - but overall there is usually only one total arrangement (or a relatively small number among all possible arrangements) that will completely meet the goal. Finding a solution within a huge range of possibilities can be very hard, and this class is often attacked using computers - but beware, the class of edge-matching puzzles is NP-Complete . You can read more on this topic in Erik and Martin Demaine's 2007 paper Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity ( download a PDF here ).

Jacques Haubrich published his definitive Compendium of Card Matching Puzzles in three volumes, in which he describes over 1000 puzzles, and a companion volume called About, Beyond, and Behind Card Matching Puzzles in which he provides interesting theoretical and historical analyses. According to Haubrich, and to Slocum and Botermans in their 1986 book Puzzles Old and New, the first edge-matching puzzle patent was applied for in 1880 by Edwin Lajette Thurston of Cleveland, Ohio (b.1857 in MA, d.1921) and granted in 1892 - see 487797 and 487798; also see 1893's 490689. Thurston was, of all things, a patent attorney! For a precursor to the edge-matching puzzle, using triangular tiles like "Triominoes," see U.S. Patent 331652 - Richards 1885. Richards doesn't claim the triangular form of the dominoes, but rather the ways to make and mark them. He never uses the word "puzzle" - he describes a number of games but none come close to being a puzzle.

In Hoffmann's 1893 Puzzles Old and New, the only edge-matching type puzzles mentioned are #72 in chapter IV "The Royal Aquarium Thirteen Puzzle" (equivalent to the French Le Nombre Treize), and #18 in chapter III "The Endless Chain" (equivalent to the French La Chaine sans Fin).

Major Percy Alexander MacMahon and his friend Major Julian R. Jocelyn applied for a patent in Great Britian in 1892 (#3297 - I could not find a copy online - it is reproduced in Haubrich's About,...) in which they describe several triangular tile games and puzzles. In his 1921 book New Mathematical Pastimes, MacMahon published some of the first material to treat edgematching puzzles with mathematical rigor. That work also discusses his 3-D puzzle Colored Cubes, first introduced in a lecture he gave in 1893.

Tile-laying puzzles (in German, "Legespiel") of both the edge-matching and polyform varieties have been explored and produced by Kate Jones at Kadon, and you can read a lot of interesting material at the Kadon site - see Edgematching Colors and Shapes, and More About Edgematching. You can see Erich Freidmann's collection here. You can see some original designs by Yukio Hirose here. Take a look at George Hart's article "A Color-Matching Dissection of the Rhombic Enneacontahedron." Also see Peter Esser's page. And Toby Gottfried's site.

Jacques suggests the following seven categories:

Jacques also defines a classification scheme by which one can identify the puzzles abstractly and find isomorphisms (i.e. equivalent puzzles).

Jacques' conclusions, based on computer analysis, regarding the best approach to solving these puzzles agree with my empirical findings - in general, fix the tile in the middle and work around it. A good tile to choose for this middle position is the tile with the most possible matches.

As discussed in my polyforms section, the only regular polygons which can be used to completely tile the plane are the equilateral triangle, the square, and the hexagon. It is evident in Haubrich's Compendium that these shapes comprise the majority of tile shapes used in existing edgematching puzzles. Rectangles have also been used, as have octagons (allowing empty areas). I've organized my example puzzles by tile shape:

Some puzzles have tiled surfaces other than a plane - e.g. the platonic solids, or a cylinder. I would file such things under 3-D edgematching puzzles.

Using this technique I can prove fairly easily that this puzzle has only one distinct solution.

There are seven hexagonal "nuts" that I label A thru G. For each nut, the six sides are numbered 1 through 6 in some order. The numbering scheme of each nut, starting with 1 and proceeding CLOCKWISE, along with the letter ID I arbitrarily assign the nut, is shown below:

All nuts must be used, each once and only once. One nut of seven must be placed in the center. In a solution, each nut except for the center must abut 3 other nuts and at each abutment the numbers assigned to the respective abutting sides must match.

Some reflection should convince you that a solution is possible if and only if one can find a set of six cells, such that:

1. each row (i.e. nut) is used once and only once
2. each column (i.e. number on the central nut) is used once and only once
3. the last number of the triple in the selected cell in a given column matches the first number of the triple in the selected cell in the next column - i.e. the numbers on abutting sides match

This is a vintage Weave-O-Gram puzzle, copyright 1951 by Edcraft-Century.   See US Patent 2439583 - awarded to Israel Shamah of London on Apr. 13, 1948.

From the patent description - the Weave-O-Gram "comprises a framework and a plurality" (patent lawyers love that term) "of flexible bands some extending in one direction and the others at right angles thereto, the various bands being interwoven... and each being capable of relative movement. On each band there is provided a number of sections of a picture... by relative adjustment of the individual bands the various sections of the picture may be juxtaposed so that a complete picture is formed. Each band will be provided with the sections of a number of different pictures so that a number of different complete pictures may be made up."

My copy allows one to make six different circus scenes. The bands move pretty freely, occasionally catching on small rough edges or tears. It is best to use both hands to move a band, slowly, from both edges of the frame simultaneously. As you might expect, Weave-O-Gram is not too challenging, however the graphics are nice and this is a great implementation of one of those ideas that seems obvious once you've seen it done.

Also see the earlier US Patent 1903226 awarded to Harry Lawson Perry of Jamaica, New York on Mar. 28 1933. Perry's device employs linear strips rather than looped bands.

Then see the later US Patent 3235262 - awarded to Ernest Frankl of Middleboro, MA on Feb. 15, 1966. It is pretty much a copy of Shamah's idea - Frankl's improvements are to "effect economies of manufacture and assembly and to provide for ease of manipulation and attractiveness." Frankl assigned his patent to Winthrop-Atkins Co. Inc. of Middleboro MA, a printing company now owned by Chilcote Co. of Ohio.

From the book Classic Century Powerboats by Frank, Trudi, and Paul Miklos copyright 2002, published by MBI Publishing Company of St. Paul MN. I learned that the company Century (as of March 2012 purchased by Allcraft Marine) makes powerboats, and was founded in Milwaukee in 1926. In 1948, F. L. "Ted" Hewitt Jr. became president. Since boat production orders were seasonal - heavy from March through July and slack during the rest of the year, one of Hewitt's initiatives was to offset the downtimes by producing other, stable products such as toys. Century produced a line of educational toys under the brand Edcraft, including Weave-O-Gram.

Shamah also patented (in the UK) a locking mechanism for a money box.

Leonard Euler (1707-1783) studied Latin Squares in the late eighteenth century, and research into them has continued, not simply because of their use as puzzles, but more for their application to experimental designs and cryptography.

The enumeration of Latin Squares has not been easy - figures up to order 10 are summarized in the table below. A reduced or standard Latin Square is one where the symbols in the first row and the first column are in lexicographical order. Given the number of reduced squares, R_n, the number of distinct squares L_n is:

L_n = n!(n-1)!R_n

In 1992, in Discrete Mathematics, J. Shao and W. Wei published a formula for the number of Latin Squares of any order. (It is non-trivial to specify.)

The Thirty-six Officers Problem goes as follows: arrange six regiments of six officers each of six different ranks into a 6x6 square so that no regiment or rank is repeated in any row or column. Note that there are no diagonal restrictions.

It turns out that there are no Graeco-Latin squares of order N=6 but this was not proven until 1901 by Gaston Tarry who exhausted all possible arrangements by hand. In 1959 Euler's conjecture was shown to be false for N > 6, by Parker, Bose, and Shrikhande. Rob Beezer shows a nice colorful order 10 square on his web page.

Since in such a superposition, the Latin Squares used cannot both be standard, a Greco-Latin Square in standard form is one where the first Latin Square is standard, and the second has only its first row in lexicographical order.

The maximum number of Latin Squares of order n which can be in a set of MOLS is n-1, but some Latin Squares have no mutually orthogonal mates.

In Greco-Latin Square puzzles of order N=4, the pieces are an assortment composed of all the combinations of two features each having four possible values. They must be arranged in a 4x4 grid such that no two with the same feature appear in any row, column, or main diagonal. Sometimes it is prohibited to have a repeated symbol among the four corners of the square, or among the four central cells (see "1000 Play Thinks" #400). There is only one order 4 Graeco-Latin Square in reduced form, but it does not meet these additional constraints. But by permuting rows, you can arrive at my solution for an order 4 Graeco-Latin Square that meets all those conditions:

Magic Squares exist for all orders except N=2. There is only one distinct Magic Square of order N=3. Supposedly the order 3 magic square was invented in China between 650 and 400B.C. and known as Lo Shu. There are 880 of order 4, and over 275 million of order 5.

In the order 3 normal square, all rows, columns, and the main diagonals total 15. The magic constant for a normal magic square of order n is given by the formula:

M_n = n(n²+1)/2

An order 4 Magic Square appears in Albrecht Durer's famous engraving called Melencolia I:

16	3	2	13
5	10	11	8
9	6	7	12
4	15	14	1

Each row, column, and main diagonal sum to 34, as do the four corners and the four central cells. Note that this is a solution to Skor Mor's Thinking Man's 34, and Reiss' 34 Skidoo of 1971.

     

Catch-21 by Gabriel (U.S. Patent 3833222 - Castanis 1974) and Hi-Q Enigma by Ideal are two puzzles where one must fit a group of pieces onto a surface bearing a pattern. Gridlock by Gabriel is a travel version of Hi-Q Enigma. The pieces match various portions of the pattern, and the proper "covering" must be found so that all the pieces will be accomodated and the surface completely covered. This will work with just patterns although both of these puzzles employ an embossed surface with corresponding holes in the pieces. Of the two, Catch-21 is more logically straightforward. It includes a set of domino-like pieces.

Here is my solution "diary" for the 2nd side puzzle of Catch-21. Please refer to the diagram at left. This shows the 6x7 board - the number inside each square indicates the pattern on the board. The numbers outside the board label the x (horizontal) and y (vertical) coordinate axes. I will refer to locations on the board via a pair of x and y coordinate pairs. The tiles consist of the 21 "dominoes" representing all ways of pairing the numbers 1 through 6.

Getting started here is not as easy as for the puzzle on the first side, since here there are multiple possible locations for every tile. Begin by considering where the 6/6 tile could be placed - there are only two possibilities: (0,2)/(0,3) or (0,1)/(0,2) shown in red. Choose to place it at the latter location. This forces 2/4 to go at (0,0)/(1,0).

Now review possible locations for the 4/6 tile - there were only two possibilities: (0,2)/(1,2) or (1,0)/(2,0) but the former has been eliminated by our choice of location for 6/6, and the latter has been eliminated by the forced placement of 2/4. Ergo we have made an error and must backtrack.

The only decision point is our choice of placement for 6/6, so we must undo everything back to that point and choose the only available alternative, placing 6/6 at (0,3)/(0,2). This still precludes one location for 4/6 and forces 4/6 to go at (1,0)/(2,0). The 2/6 tile must go at (0,0)/(0,1).

Now consider the 6 at location (5,0). It could be covered by either the 2/6 tile or the 3/6 tile, but 2/6 has been used so 3/6 must go at (5,1)/(5,0). Next consider the 3 at location (0,6). It could be covered by the 3/6 tile or the 3/4 tile, but 3/6 has been used so 3/4 must go at (0,6)/(0,5).

All squares in column 0 have been covered except for (0,4), which now must be covered using the 1/4 tile at (0,4)/(1,4). Now look at the 6 at location (1,6). It could be covered by the 2/6 tile or the 5/6 tile, but 2/6 has been used so the 5/6 tile must go at (1,6)/(2,6). This forces us to place the 2/4 tile at (1,5)/(2,5). Now consider the 6 at location (5,4). It could be covered by 1/6, 3/6, or 5/6, but we have used the latter two so it must be 1/6 at (4,4)/(5,4). Now consider the 4/4 tile. The only remaining possible location for it is at (2,4)/(3,4). Likewise consider the 1/1 tile. It can only go at (3,3)/(3,2).

Next consider the 5 at location (4,3). Either way it must be covered by the 5/5 tile. This implies that the 1/5 tile must be aligned next to it. There are two ways of placing them and either is OK.

Consider the 1 at location (4,1). Either way, it must be covered by the 1/2 tile. This implies that the 2/5 tile must be aligned next to it. There are two ways of placing them and either is OK.

Now look at the 2 at location (2,1). Since the 2/5 tile is used, we must use the 2/3 tile here. Next consider the 4 at (1,2). Either way it must be covered by the 4/5 tile and the 3/5 must be aligned next to it. There are two ways of placing them and either is OK.

There remains only 1 location for 2/2, at (3,6)/(3,5). Lastly, consider the 3 at (5,6). It must be covered by the 3/3 tile and the 1/3 must be aligned next to it. Again, two arrangements are possible and either is OK.

You can make and play this type of puzzle with just some paper and a set of dominoes. Wonder Workshops had this inexpensive version of a domino puzzle (I don't have it). An arrangement of pips is presented on a card. Using a set of dominoes, cover the card by matching pairs.

Dominola - created by Eric Everett ca. 1930's, issued by Traydac
A set 28 of standard dominoes and a two-challenge board.
See a nice writeup of Dominola by Mike Desilets at Kevin Sadler's blog

Other good resources for Domino Puzzles:

• Jurgen Koller's page
• Domino Logic
• Frans Faase's page

Instant Insanity was invented by Franz O. (Frank) Armbruster, a California computer programmer, and marketed by Parker Brothers during 1966-67. It sold about 12 million copies.

I have an original from Parker Brothers, and copies from Winning Moves (found in a shop in Mystic) and Kadon.

There are many puzzles in what I call the "Instant Insanity Family." Typically, the puzzle comprises four colored cubes. Four colors are used, and each of the cubes' faces are colored with one of the four colors. The objective is to find a linear arrangement of the cubes (a rectangular prism) such that all four long sides show each color only once. The faces on the ends and in between cubes don't matter. There are 41,472 distinct ways to line up the four cubes, so exploring them all by hand will take a while!

The number of ways to color a cube, allowing repeated colors, with at most k colors is given by Polya's Enumeration Formula, which follows from Burnside's Lemma. The formula for a cube is:
n = (k⁶ + 3k⁴ + 12k³ + 8k²)/24

This gives the following values for k=2,3,4,... (Sloane's integer sequence A047780):

10, 57, 240, 800, 2226, 5390, 11712, 23355, 43450...

So for four-color type puzzles such as Instant Insanity, there are 240 possible cubes to choose from when creating the puzzle. The trick is to find a set that guarantees a solution, and preferably only one solution!

The cubes are typically diagrammed by "unfolding" them and representing them as a cross:

Note that when coloring a cube, once you've colored two pairs of opposing faces, then given colors for the last pair of opposing faces, you define a pair of mirror image cubes by assigning the colors to those two remaining faces in one way or the other. For purposes of this type of puzzle, mirror image cubes are identical and distinctions between them are immaterial to the solution.

Ivars Peterson's MathTrek article of August 9, 1999, "Averting Instant Insanity," is devoted to graphical solution techniques pertaining to this type of puzzle.

I learned from David Singmaster's notes that, according to Thomas O'Beirne, writing in the Mathematical Gazette, Vol. 41, No. 338, Dec. 1957, the graphical technique seems to have first been published in 1947 by Cedric A. B. Smith, writing as "F. de Carteblanche" in Eureka No. 9 (April 1947) - "The Coloured Cubes Problem" pp.9-11. O'Beirne discusses the puzzle in his 1965 book Puzzles & Paradoxes in chapter 7, though he doesn't reference Carteblanche there. O'Beirne's article on the "Great Tantalizer" from his Puzzles & Paradoxes column in the New Scientist of Aug. 10 1961 is online at Google books.

For the graphical solution technique, the set of cubes is represented as a single undirected pseudograph. The term "graph" usually formally excludes more than one edge connecting any two nodes, and self loops. The term "pseudograph" allows both, and we need both for this type of puzzle. Each of the four colors is a node. Each cube results in three labeled edges on the graph (for twelve edges altogether) - an edge connecting two nodes (or a node with itself) when those two colors appear on opposite faces of the cube, labeled with an identifier for that cube. Given this graph, the solution to the puzzle can be determined, and all puzzles of this type with isomorphic graphs are in essence identical. Note that the Graph Isomorphism Problem is a well-known area of research.

On the right is the graph for Instant Insanity (I numbered the cubes arbitrarily).

Recall that there are 240 ways to color a cube using 4 colors. Given a "stable" of 240 cubes from which to choose four, how many possible 4-cube sets are there? To allow repeats but eliminate sets that are identical (since the ordering of the 4 cubes does not matter), we use the formula for a multicombination:

(n+k-1)_C_k

In our case, n=240 and k=4, (240+4-1)_C_4 = 141,722,460. So, we've got over 140 million sets to explore! Also, for a set of 4 cubes, there are 41,472 possible ways to arrange them - that's going to be a lot of work.

But we can reduce this by eliminating from consideration all cubes that do not have at least one face of each color. It's a little arbitrary, and it means my conclusions are not universal over the entire universe of possible puzzles, but I think it's reasonable - most commercial puzzles comply. Of the 240 possible four-color cubes, there are 68 that use all four colors at least once each.

Each cube can be represented by a four-node, 3-edge pseudograph, where the nodes are the four colors symbolized by the letters ABCD and each edge corresponds to a pair of opposing faces on the cube. The graphs are the same for mirror-image cubes, since the edges are undirected. The 68 cubes give rise to only 52 unique graphs. Those 52 graphs can be grouped into 6 groups - each group is closed with respect to a permutation of the node/color assignments, of which there are 24 (4!).

Below is a chart I made identifying the 52 graphs/cubes by unique ID number (00-51) and a small diagram showing the 3 edges. The colors identify the six groups. It should be easy to construct a cube given its little diagram and the key on the lower right. You just have to decide on your assignment of actual colors to ABCD, and then be sure to remain consistent.

Rick Eason worked on this problem, too, after we talked at NYPP2010.
Rick assigned numbers to these groups as follows, and noted that cubes in a group have the given number of possible effective orientations within a puzzle, ranging from 10 to 24.
A higher number of possible orientations might contribute to a harder puzzle.

yellow (24) green (16) teal (16)

lavender (24) tan (10) blue (12)

---

A children's construction toy called Clics, can be used to make various 4-color cubes to explore Instant Insanity type puzzles.
You can see some of the cube models I made - I affixed a small label to each cube, showing its ID number.
One bucket does not contain enough color panels to make all 52 cubes simultaneously.

Here is a chart of number of solutions, number of sets that have the given number of solutions, and the first set found that has that number of solutions. I have omitted solution counts where no set had that number of solutions.

A set is represented by its "signature" - an eight-digit number (zeroes on the left, if dropped, should be inferred) that is simply the four cube IDs (00-51) from the chart, concatenated, always from lowest numerical ID on the left to highest on the right.

0)	132647	00010203
1)	5160	00010218
2)	21186	00010234
3)	3428	00010329
4)	38466	00010318
5)	2088	00011839
6)	8070	00011631
7)	612	00031631
8)	28626	00010518
9)	702	00032939
10)	3276	00012039
11)	282	00031839
12)	7310	00011639
13)	312	00032637
14)	1380	00032651
15)	144	00033539

16)	9060	00012750
17)	120	00033139
18)	360	00032039
20)	1980	00032549
22)	111	00052039
23)	12	03082645
24)	2340	00032338
26)	204	00162238
27)	1	00193141
28)	486	00033847
29)	18	00194045
30)	30	00163147
32)	1188	00051639
34)	18	01163246
36)	114	01033839

40)	420	00163841
44)	156	01030539
48)	94	00274250
52)	12	01164045
53)	3	06184045
54)	6	01283844
56)	96	01063847
60)	6	03113949
64)	120	01030523
72)	51	01032338
80)	3	24274350
96)	12	01233438
128)	6	03052339
140)	3	01163847
160)	6	01052338

I found 5160 sets that have a single solution. There were many sets - almost half - that had no solution at all. Some sets have more than one solution, and the maximum number of solutions any set can have seems to be 160. Most solvable sets have four solutions. There is only one set that has exactly 27 solutions, and 27 solutions is the only count where only one set has that count. Interestingly, the cubes used for this set are the four from the yellow group in the chart. It is not hard to solve this set by hand.

Based on a cursory check, it appears that none of the cubes in groups 5 (tan) or 6 (blue) are used at all in single-solution sets!

Rick Eason says that by using Burnside's Lemma to eliminate color permutations, the 270,725 figure can be collapsed to 11,746 sets. If we further eliminate the 12 cubes not used in single-solution sets (the tan and blue groups), we end up with only 3949 sets. I did not perform this collapsing for my runs, so my totals do not take into account color permutations.

In a brief 1977 paper entitled On "The Tantalizer" and "Instant Insanity" (PDF online here), Frank Harary (1921-2005), recognized as one of the fathers of modern graph theory, notes that "Using standard methods of graphical enumeration [Harary and Palmer 1973], it is not difficult to develop a formula for the number of different 'Tantalizer' games with four cubes. This can be done by associating a graph with each such game, and then counting graphs." When I first began to research this topic, I checked Harary's book out of the library, but I failed to find a handy formula for enumerating these graphs. If you know of one, please let me know!

The set identified by O'Beirne in his Great Tantalizer article in Figure 1 is the first listed, 01042040.

SK consists of 1 cube from group 2 (green), 1 cube from group 3 (teal), and 2 from group 4 (lavender). Rick suggests that the most difficult puzzles will consist of only cubes from groups 1 (yellow) and 4 (lavender), since those groups offer the most possible orientations (24 each). Rick found the following 13 single-solution sets using only cubes from groups 1 and 4:

00 03 04 19
00 03 04 20
00 03 04 30
00 03 04 31

00 03 04 39
00 03 14 19
00 03 14 20
00 03 14 22
00 03 14 37

00 03 14 39
00 03 14 41
03 04 08 12
03 04 12 20

The certificate with the smallest number of sets having that certificate is shared by only six sets:
02202240022022402240224002202240
02202240224022400220224002202240
22402240022022400220224022402240
is assigned to sets: 00030419, 00071231, 00081441, 19203031, 19223741, and 31333941.

In each set, a given pair of colors appears only once each on adjacent sides on all four cubes, with the other four sides of each cube having the other two colors. The graph of such a cube will have only one edge leading from each of the two colors, which will not share an edge. Four colors, taken two at a time, 4_C_2, gives the 6 possibilities. In signature order, they are: CD, BD, BC, AD, AC, and AB. These sets are easy to solve - identify the special colors. Orient all cubes so that the sides with the special colors are on top and front. Now it is a simple matter to turn three of them so that the solution is found.

At the other end of the scale, there is one certificate that encompasses 144 sets:
11201121112112211221122111201221
11211121122112211120122111211221
12211221112012211121122112211221

The first example having that certificate is 00162936.

According to my analysis, there should be 200 distinct single-solution puzzles, where every cube uses all four colors, and each set uses four distinct cubes. You can download a text file "single-soln-sigs-and-certs.txt" in which my program output cube set signatures and certificates for every single-solution set. Those sets isomorphic to SK are noted. The tail of the file contains a summary of the 200 unique certificates and the count of sets having a given certificate.

The table below lists, in (mostly) alphabetical order (I tried to keep sequels together), those I know of and shows images where I either have a copy or have been able to find pictures. I don't have them all and I will note the ones I don't have. Those I have are highlighted like this. Those I do not have are highlighted like this.

I would like to acknowledge the following:

• The generous Liz Burrow, who gave me several of these puzzles, and Martin Watson, who was kind enough to bring them to me in Prague.
• Colin James, a fellow collector, who has also sent me puzzles and helped catalogue the puzzles, supplying many photos of his collection.
• Several photos of items I don't have are from the Slocum collection courtesy, The Lilly Library, Indiana University, Bloomington, Indiana.

Various pamphlets or sheets accompanying the different puzzles mention other puzzles in the lineup. Several mention 5 puzzles but list 6; one describes Mindbenders as "a fiendish series of six diabolical puzzles" but then lists eight. I have found ten altogether.

If each of the n faces of a regular solid is colored with one of n colors, the number of different ways the solid can be colored with all n colors excluding rotations but including reflections is given by n!/(2*E) where E is the number of edges of the solid. For cubes with 6 faces and 12 edges, the number of possible colorings is 6!/(2*12) = 720/24 = 30. These 30 cubes form the set of MacMahon Colored Cubes.

You can choose any of the 30 cubes to use as a "prototype" and it will be possible to find eight other cubes in the set which can be used to build a 2x2x2 model of the prototype having the same arrangement of solid colors on the six external 2x2 faces as on the prototype's six faces, and also satisfying the additional constraint that internal touching faces are colored alike. MacMahon credited his friend Colonel Julian R. Jocelyn with the discovery that this can always be done regardless of which of the 30 cubes is chosen as the prototype.

For each prototype, there is only one set of eight cubes which will work, and there will always be two ways to build the prototype with these eight cubes. A procedure to transform one solution into the other was devised by L. Vosburgh Lyons and is shown in [2] on page 190. The eight cubes to be selected will not possess any of the same pairs of opposing face colors as the prototype. This means that 13 of the 29 cubes can be eliminated as candidates, leaving 16 from which to choose.

Lyons also discovered that after a 1st prototype is selected along with the eight cubes to model it, it is always possible to select another prototype from the remaining 21 cubes, and find another 8 cubes from the remaining 20 to model this 2nd prototype. The 2nd prototype must be a mirror image of the 1st, and the eight cubes to model it are the eight not chosen from the 16 eligible for the modeling of the 1st prototype [2].

MacMahon sold an eight-cube puzzle patented in 1892 (UK 8275?) to the London company R. Journet, which marketed it as the Mayblox puzzle [4]. Its eight cubes are one of the sets which can model a prototype and meet the internal color-matching constraint. However, the Mayblox puzzle does not specify the configuration of the prototype - so it must be deduced, which makes the puzzle more difficult.

I made my own version of a Mayblox puzzle using LiveCube. I chose a prototype arrangement of six colors at random, then colored the 8 sub-cubes as required based on the solution instructions in [1]. I finally found a good use for the LiveCube face plates! Fortunately, they provide enough different colors with the addition of cyan and pink to their previously available black, red, yellow, green, and blue.

Odd Man Out is a version of the Mayblox puzzle, produced in wood using numbers rather than colors, by Dave Janelle at Creative Crafthouse.

You are to build a 2x2x2 cube using 8 of the 9 included numbered dice, so that every face displays a unique number.

1. Mathematical Recreations and Essays, by W.W. Rouse Ball, Macmillan 11th ed. 6th printing 1973, pp.112-113
2. Torsten Sillke provides a good bibliography which led me to other sources, including the Martin Gardner books I already have but hadn't realized contained pertinent information
3. New Mathematical Diversions from Scientific American, by Martin Gardner, 1966 Simon and Schuster, pp.184-195
4. Fractal Music, Hypercards, and More, by Martin Gardner, 1992 W.H. Freeman and Co., pp.88-99; discusses the Conway matrix
5. my section on Journet
6. Juergen Koeller's site
7. MacMahon bio