Pattern Puzzles
             

Pattern puzzles are a sub-class of Put-Together puzzles. They consist of a number of similar pieces or movable parts, each of which has some particular identifying trait or traits whose values are chosen from a small well-defined set - sometimes shape, but more often a color or pattern, along the edges or on the face(s) of the piece.

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 ).

Edge Matching

Edge-Matching puzzles usually consist of a set of tiles whose edges have various distinct patterns, symbols, or colors. The objective is to arrange the tiles in a grid such that abutting edges "match" according to some rule. Since the tiles can be printed on cardboard, these puzzles can be inexpensive to produce, and in the past have served extensively as advertising promotions and giveaways. There are many modern edgematching puzzles - you may have heard of "Scramble Squares" for example.

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:

  • EM - Edge Matching
  • HT - Heads/Tails - an elaboration of the edge-matching theme - instead of matching equals, one must match corresponding pairs.
  • CP - Continuous Path (I have categorized these under Route Building)
  • CM - Corner Matching
  • CD - Corner Dismatching (e.g. Grandpa's Wonder Soap)
  • JS - Jigsaw-type (each tile's edge has only one possible mate)
  • HY - Hybrid
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:

[Square Tiles] [Hexagonal Tiles] [Triangular Tiles] [Other Tiles]

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.

Square Tiles


The Thurston design was used by Calumet ca. 1930 as advertising for their products. In the Calumet Puzzle, one must match not just the proper color edges, but also the corresponding top and bottom halves of the baking powder cans. I solved the Calumet puzzle using a variant of my Drive Ya Nuts technique.

The Grandpa's Wonder Soap Puzzle was patented (U.S. Patent 1006878) in 1911 by A.K. Rankin. It is described on page 36 of Slocum and Botermans' "Puzzles Old & New." The goal is to form a 3x3 grid such that at the points where the quarter-circles on the corners of the tiles meet (either four or two), there are always different colors on each of the meeting quarter-circles. Grandpa's head must always be upright on every card, so the cards cannot be rotated. Jacques says this is the first example of a corner dismatching puzzle. There are different versions.

The Besco Soap Puzzle - from the Beaver Soap Company of Dayton Ohio. Form a 3x3 grid such that there are four different colors in each circle at the corners, and two different colors in each half-circle. There must also be a different color at each of the four corners of the grid. The pieces must all remain oriented with the words "Besco Soap" upright.
JH Vol.2 p163- 2 solns

Le Berger Malin - The Lazy Shepherd (N.K. Atlas Paris circa 1910). NOTE: "malin" can also be translated as "ingenious." There are nine tiles, each of which is divided into four quadrants by two diagonals. There is a number in each quadrant, and the corresponding number of sheep are depicted. Arrange the tiles in a 3x3 grid such that the total in each square formed where edges meet totals 10. According to Jacques, this is the first known example of a Heads/Tails puzzle (the two numbers which must add to ten comprising the head and tail).

Le Fermier Avise - The Wise Farmer
The same principle as the Lazy Shepherd, except with chickens and now it's a wise farmer.

This is a French puzzle (Casse-Tete) called "Le Nombre Treize" - The Number Thirteen. Arrange the tiles in a 3x3 square such that the red figures align vertically and the blue figures align horizontally, and such that the three figures in each of the six rows and six columns add to 13.

The Vess Cola Nine Piece Puzzle is an advertising giveaway, promoting Vess Cola; its pieces are equivalent to the Calumet Puzzle.

The Vess Mystery Puzzle
two versions

Here is a 3x3 heads/tails that Norman Sandfield found on a visit to the remote Easter Island (Rapa Nui). He bought all he could find, and I got one from him at the 2006 New York Puzzle Party.

Pepsi edgematching puzzle (Spanish)

La Canadienne - an advertising promotion edgematching puzzle

The "33 to 1" Puzzle, advertising Pabst beer. Copyright 1940. One version shows bottles, the other cans.

Pabst 33 - arrange the cards in a 3x3 grid so that the numbers total 33 horizontally, vertically, and along the main diagonals.

Pel-Freeze
JH Vol.1 p93 - two solns

A-Treat Mystery Puzzle - A-Treat Bottling Co. PA
Three versions.
Jacques' Compendium, Volume 1, pp72-4, indicates that the following 9-square puzzles are isomorphic and have only one solution:
  • Le Berger Malin
  • Le Fermier Avise
  • The Calumet Puzzle
  • Pabst 33 to 1 cans
  • Pabst 33 to 1 bottles
  • The Vess Cola Mystery Puzzle (bottles)
  • Vess Cola Mystery Puzzle (caps - violet,brown,green,red)
  • Vess Mystery Puzzle (caps - violet,orange,blue,red)
  • A-Treat Mystery Puzzle

Nestle Nine-Square

Lionstone Whisky, Kentucky 1973

Camel - an advertising promo from a cigarette manufacturer.

Bits and Pieces - match the color dots on the squares' edges

Nitty Gritty - Arrange the colored tiles in the tray so that the edge patterns on all adjacent tiles match.

In Mattel's Mangle Quadrangle one also has to arrange the colored tiles in the tray so that the edge patterns on all adjacent tiles match. One in a series of Brain Drain puzzles.


Loncraine Broxton/Lagoon Games offers a set of four Professor McBrainy's Zany optical illusion puzzles: Fusion, Cosmos, Vortex, and Kaleidoscope. The patterns are psychedelic.

Ovals - Nob - O-Pico/Color Match
Puzzland Hikimi
JH Vol.1 p151 - 6 solns

Chelona Pocket Puzzle - Cats
A nine-square edge matching puzzle from Chelona in Greece. Cute, colorful graphics on thin plywood. Novel packaging - an extra backing tile has an attached elastic cord to enclose the stacked tiles. A solution diagram is on the bottom. Available from Padilly.

Crafty Butterfly

IZZI - Binary Arts
Geo Matrix made by Binary Arts for The Museum Company is the same.
(I don't have Geomatrix.)
JH Vol.2 p303

Zoki
See U.S. Patent 5524898 - Pavlovic 1996

At the January 2007 Toy Fair, Tomy launched the Eternity II puzzle (Wikipedia entry here) and a contest prize of $2M USD. Eternity II employs 256 square tiles to be arranged in a 16x16 grid, which must have a gray border and a particular tile at a given "starting" position. The "final scrutiny date" passed without a winner, so the prize went unclaimed.

Scramble Squares
Tropical Fish
Sea Shells
There are many different versions of Scramble Squares, and most of them are in fact distinct puzzles (not just the same puzzle with different pictures).

12 Ladies, from the Dr. Wood Mind Challenge series.

A Tens Edgematching puzzle from the Plantecs at PuzzleMePlease.com. Same idea as Le Berger Malin.

One Tough Puzzle is a 3x3 heads/tails edge-matching puzzle. I solved it using an extension of the technique I describe for the Drive Ya Nuts.
JH Vol.1 p107

Another Tough Puzzle is a 7-piece hexagonal heads/tails edge-matching puzzle.
(I have listed it here with the original One Tough Puzzle.)
JH Vol.1 p35 - 1 soln

Another Tough Puzzle (Triangles) - Great American Puzzle Factory
  
The Ultimate Puzzle (4x4 square tiles)
The Ultimate Puzzle Two (3x3 right triangle tiles)
Both designed by Lee Willcott of Estonia.
The original version has square tiles - each side has a cutout or tab, in one of four shapes: an "in" arrow, an "out" arrow, a cross, or an oval.
There are four challenges: build a 2x2; build a 3x3 (find 9 tiles that work); build a 4x4 with no side-up constraint; build a 4x4 with either all smooth sides or all rough sides up.

The Ultimate Puzzle Two has 18 right-triangular tiles, pairs of which form squares. Nine squares must be arranged in a 3x3 grid (so I have listed it here in the square tiles section with the original version). Each tile has a cutout on each side - the cutouts form halves of four symbols: crown, heart, spade, and star. On one pair of tiles, a heart on the diagonal is offset - the bumps half is lower and the v half is deeper. Where tiles touch the halves must form a whole symbol, so this is a "heads and tails" type edgematching puzzle.


DaMert 3D Squares Cars

4D Metapuzzle

4D Metapuzzle No. 4

Instant Insanity - Hexagon Puzzles 1986
JH Vol.1 p151 - 2 solns

A four-piece glass edge-matching puzzle made by Brett.

The Great Canadian Puzzle

Setko Match Heads

Peterson Uptight Spider
JH Vol.1 p149 - 1316 solns

The Wobbly Web
Create a rectangle from the 15 square tiles such that web strands join (edgematching).
JH Vol.2 p210
(See the Waddington's series, below.)

Peek-a-Boo Snakes

Eye-Cue

Pin Pin Kan

New Departure Puzzle - advertising New Departure coaster brakes for bicycles. An uncut card. Copyright 1953.

Celtic Knotwork

Lagoon Group Safari series - Tiger Puzzle

Warning - Lagoon

Jungle Mega Puzzle - Lagoon

Vintage Chevrolet advertising edgematch.

"Ladybugs," a "3D Magna Puzzle"
by Caeco, purchased from New England Hobby.
16 magnetic square tiles. A holographic image that is 3D when viewed from two orientations, and flat from two others. This is more of a jigsaw since each edge really has only one mate.

A British Petroleum (BP) advertising edgematching puzzle from Australia.

The Crazy Cheese Puzzle, from Blue Orange Games. A nice wooden 3x3 edge-matching puzzle with two levels of challenge.
Purchased at The Games People Play.

Holland America Edgematching puzzle

Suits Edgematch
I found this at Der Verrückte Laden, a puzzle shop in Berlin.

Perfect 10
from Creative Crafthouse
A vintage advertising cardboard edgematching challenge, the Snyder Standard Nine Piece Puzzle. According to the package, "Par is tweleve minutes."

 

 

This vintage six-piece, square-tile edgematching puzzle from Germany is called the Matador Dominoes-Puzzle.
The backs of the tiles are blank. I have solved it, so it can be done.


Fish Frenzy - arrange the eight cards to form a picture, then move the cards about to form other pictures.

Hexagonal Tiles


The Daily Mail World Record Net Sale puzzle was published in 1920 and has only one solution. See British Patent No. 173588 - Hydes and Whitehouse 1921. It was printed and published for Associated Newspapers, Ltd. by the Chad Valley Co. Ltd of Harborne, England. It consists of 19 hexagonal tiles which must be arranged in a large hexagonal grid 3 tiles on a side, while matching the color and the word on adjacent tile sides. The newspaper ran a contest and offered various prizes, and my copy came with a yellowed newspaper clipping showing the winner - one Mr. C. Lewis of Dalston - and the solution.
JH Vol.2 p246

The Nestle's puzzle, published around 1930-1940, was the first 7-tile hexagonal edgematching puzzle. I found one from Malaysia, and one printed in Spanish. There have been many variants, including the modern "Drive Ya Nuts."
JH Vol.1 p30 - 1 soln

To me, the edge-matching puzzle is exemplified by the more recent but none-the-less venerable Drive Ya Nuts. I developed a tabular solution, shown below. There was also a version with red nuts in a silver case - it has the same numbering.

The Circus Seven by Masudaya is fairly well-known. In principle it is the same as Drive Ya Nuts, but with tents and colors instead of nuts and numbers. I solved mine easily using my tabular technique.

The Circus Puzzler is a clone - many variations exist.

Dunlop Hexagons

Lyons' Tea

Match the Colors - Adams ca. 1953

Gemstone - Nordevco

Pressman's Think-Ominos

Melissa & Doug make a similar puzzle called the "Space Edge Matching" brain-teaser

Spider Puzzle - Toysmith

Hexamania
From The Long Point Toy Co. of Ingersoll, Ontario.
19 single-sided hexagonal tiles numbered 1 through 19, each having the edges differently numbered 2 through 7. Arrange the pieces in an edge-3 hexagonal grid (i.e. 5 rows - 3/4/5/4/3) such that all adjacent edges total 9. A heads/tails type of edgematching puzzle.

Another Tough Puzzle is a 7-piece hexagonal heads/tails edge-matching puzzle.
(I have also listed it with the original One Tough Puzzle.)
JH Vol.1 p35 - 1 soln

Pair It - made by Steve Kelsey
Arrange the seven hexagonal pieces such that edges match.

Triangular Tiles


The OXO Triangle puzzle. This was published in 1922 by the OXO Company for a contest. There are 25 equilateral-triangular cards/tiles, numbered 1 through 25. The tiles must be arranged into a side-5 large triangle such that at each point where the corners of different tiles meet, the colors are different, and such that at each point where three tiles meet they spell "OXO" and where six tiles meet they spell "OXOXOX." There are 104,920 solutions but finding even one by hand is very difficult.
JH Vol.2 p271

Tri-Puzzle - Lagoon

Contack
A game (not really a puzzle), played with 36 triangular pieces. The 3 sides of each triangle are different in colour, and each side has a number from 1 to 10 printed at its edge. Players try, by matching color and number, or matching color and making certain score totals, to play all the triangles from their hand and to make as large a score as possible. Listed at Boardgamegeek.

Butterflies Mini

Travel Triazzle

12 Triangles - Majak

Playtime Flower Puzzle
General Toys No. 210 Hong Kong

BrainArt Level 4 - Ekos
Designed by Klaus Schröer
Arrange the 24 equilateral triangular tiles into a hexagon with one lizard in each cube, then solve four riddles.

BrainArt Infinity - Ekos
Designed by Klaus Schröer
Arrange the 24 equilateral triangular tiles into a hexagon so that only one continuous snake is formed, then solve a hidden mystery.

Jinxed, a vintage edgematching puzzle by Tryne.
There are 24 triangular tiles, the edges of which are colored with a dot in one of four colors (red, yellow, white, and green). The tiles are to be arranged into a hexagon such that all touching edges match and the border is all red. (It is possible to solve the puzzle with other border constraints, too.) Puzzles using triangular tiles are discussed by MacMahon in New Mathematical Pastimes on pages 2-22. MacMahon shows the possible tile sets for 1, 2, and 4 colors in Figure 2, and gives a solution for the 4-color set.
Kadon's Multimatch III is essentially the same puzzle.

Another Tough Puzzle (Triangles) - Great American Puzzle Factory
(Also listed with the original One Tough Puzzle that has square tiles.)

Triazzle - Alice in Wonderland
 

Other Tiles


This is L'Arc en Ciel (The Arc in the Sky - i.e. the Rainbow). There are nine discs on pegs, arranged in a 3x3 grid of holes in a baseboard. Each disc has four diamonds arranged pointing north, south, east, and west. Each diamond is one of eight colors: red, green, yellow, orange, gray, white, pink, and blue. The discs must be arranged such that no color appears more than once in each of the three rows and three columns of six diamonds. In this case, not every color appears in every row or column. There is a second constraint - when a pair of colors is placed adjacent horizontally, then that same pair cannot be placed adjacent again in that direction - likewise for the vertical.
JH Vol.1 p162 - 236,326 solns

  
Tile-O-Rama
Called the "Wonder Mosaic Puzzle" when it first appeared circa 1925, this was the first design with rectangular tiles. (I have a loose copy but it was also included in the F.A.O. Schwartz Deluxe Puzzle Chest No. 3006.)
I also have the Mosaik and Tesa Mosaic Square puzzles discussed in Slocum and Botermans' "The Book of Ingenious & Diabolical Puzzles" on p.23.


Here is a new favorite, an RGB Roundup from Elverson. The basic puzzle is to build a circle in the provided tray, such that adjoining colored dots match. Additional objectives include building various shapes like in Tangrams. The pieces are made of very nice weight plastic material like Mah-Jong tiles.

Vintage Set of Four Colorful Circular Torture Puzzles by Shackman

IZZI 2 - Binary Arts
Color Matrix made by Binary Arts for The Museum Company is the same.
JH Vol.2 p198

Domino Match

Double Half - Interkemia 1981

Block Shock

Ed Pegg, Jr. developed the game Chaos Tiles based on a non-periodic tiling. I've read that Ed has yet to determine if the shapes can in fact completely tile the plane.

Around the Barn

Concentra Atlantis Dan Gilbert 1994
A circular edgematching puzzle with pentagonal pieces.
Not too difficult since the die cuts give clues.
 

3D Edge-Matching Puzzles

These three puzzles, Nice Cubes, Mental Blocks, and Double Disaster, each comprise a set of transparent cubes with colored markings. Their goals, respectively, are: to line up four cubes so all four long sides each show all four colors; to create a cube so that embedded colored rods match where the pieces touch; and to create a cube such that each side shows all four colors, then such that each side shows a solid color. Double Disaster was offered by Scientific Games Division of KMS Industries Inc. of Ann Arbor Michigan, and its insert mentions the other two. Mental Blocks was offered by Creative Playthings.
The Nice Cubes in their original package say Copyright 1968 Funtastic, Div. of KMS Industries, Inc., Alexandria, VA.

   


This is Aquarium designed by Kohfuh Satoh.
I bought it at Torito.

Three challenges - no incomplete fish ever allowed to show, but unless otherwise noted fish may span edges/turn corners:
1. Use 3 pieces and build a 1x2x3 showing 7 fish.
2. Use all 4 pieces and build a 2x2x2 showing 9 fish. It is also possible to show only 8 or 7.
3. Use all 4 pieces and build a 2x2x2 showing only 6 fish with none turning any corner.

The Dodeca Nona puzzle comprises a dodecahedral magnetic body, and 12 2-sided pentagonal tiles. The tiles are numbered 1-5 at their vertices in all possible orderings. The objective is to arrange the tiles around the dodecahedron so that the 3 numbers that meet at each vertex add up to nine.
JH Vol.2 p202

Integr8, from the Dr. Wood Mind Challenge series.
Arrange the 8 cubes into a larger cube such that only complete circles show on the sides - no semicircles or lines can be visible.
Here is a solution for Integr8:

 


This puzzle, called Crux, was made (in Hong Kong) back in 1972 by the Hubley Division of Gabriel Industries - it's marked "Copyright 1972 Gabriel Ind. Inc." It is a version of the Piet Hein design Triple Cross (that I do not own). It consists of six plastic pieces, each with four color spots on the sides. Twist the blocks to ensure that at each intersection, there are three different (or same) colored spots. Pieces are paired by elastic cords threaded through the narrow end and anchored inside on hooks molded on the back plates. The elastic cords had long since worn out, but you can easily pop out the back plate by poking an unbent paper clip through the hole. I replaced the cords with rubber bands. If any of the internal plastic hooks are broken, just anchor the band with a paper clip.
In this group of puzzles, several rods have been cut into fragments of different lengths - these must be fit into a base containing sockets drilled to different depths, such that the completed stacks all achieve the same height.

In Reiss' Flat Top, arrange the eight pegs in the four holes so that each stack ends up at the same height.

In Kohner's Even Steven, match pegs to sleeves such that all pegs align at the same height.
U.S. Patent 3375009 - Stubbmann 1968

Here is another version of Even Steven, with blue plastic rod pieces to be inserted into a clear base. Really closer to Flat Top.

Tekozuru Z2 from Hikimi - purchased from Torito

Adele and Peter Plantec at Puzzle Me Please sent me a selection of their beautiful puzzles crafted from exotic woods. This one is called Double Dowel. Thanks!

Mental Misery (aka Double Trouble) - Lakeside
A transparent box, a frame which fits inside the box and will hold four cards vertically against the box sides, and five cards each colored with four colors front and back. The four colors are red, yellow, green, and blue. Arrange the cards on five sides of the cube so that edges match inside and outside the cube.
The instructions are marked: Made in Hong Kong, Lakeside Industries, a division of Liesure Dynamics, Inc. Minneapolis Minnesota. Copyright 1970 Leisure Dynamics, Inc. The box is marked: Copyright 1969 L.I.I. Made in Hong Kong. The frame is marked: Copyright 1969 Lakeside Toy Division of Lakeside Industries Inc. Made in Hong Kong.

The Great Pyramid Pocket Puzzle, by Eliot Inventions Wales 1981, is a tetrahedron with 4 equilateral triangular tiles pegged to each side. The triangles are printed on one side only with a series of radiating wedges of different widths. The objective is to arrange the triangles so the edge patterns on all adjacent triangles match. The larger version with 9 triangles per side carried a 25,000 GBP prize for the first solver. I have no idea if it was ever awarded. The small case is a mini-puzzle in itself - a 2-piece trick box.

Another oldie but goodie is On-the-level by Mag Nif.
Fit 9 multi-level pieces into a 3x3 grid such that wherever 2 pieces meet along an edge it is at the same level. In addition, the solution must be "toroidal" - i.e. opposite outside edges must all also match.
JH Vol.1 p161

In Contoura, arrange the blocks so that the surface contour is correct.

Four Cube Impossible Puzzle - Binary Arts 1998
Created by Scott Nelson in 1971 - maps the 24 3-color MacMahon squares to the faces of four cubes such that all edges match.
The puzzle gives 13 arrangements to make and requires that all adjoining edges match.
Analyzed by Jaap.

Chain of Colors - Haubrich IPP26

Ey-Ko-Se
Imperial Games Ltd. Southport England
Edgematching triangular tiles on the faces of an icosahedron. Alternative challenges include total 15 on every pentagon of five triangular faces, or show a 5-letter word on every pentagon.

Einstein Cube

Dodek-Duo
Issued by the Dodek Puzzle Company (dead link?)
Designed by Jerry Langin-Hooper
12 magnetic pentagonal pyramid pieces form a dodecahedron. The edges of each pentagonal face have a color (one of five), and a small black shape (one of rectangle, semicircle, triangle). Try to assemble the dodecahedron such that the colors match at face edges (only one solution), then such that the small shapes match (47 solutions, but much more difficult to find). It isn't possible to do both simultaneously.
Analyzed by Jaap.

Dodecahedral edgematching - R.D. Rose
Purchased from Norman Sandfield at IPP 29 in SF.

The Enigma (a dodecahedral edge-matching puzzle) by Reiss, 1971

Colin James sent me a rare Twitchit puzzle, issued in 1972 by the Hubley Division of Gabriel Industries of Lancaster PA, and designed by Piet Hein. This is a dodecahedron whose faces turn. One of three symbols (a circle, square, or triangle) is printed at every vertex of every pentagonal face. The objective of the puzzle is to rotate the faces (in place) so that at all 20 vertices of the dodecahedron, three different symbols show. Although discolored with age, the plastic remains intact and the puzzle is playable. I solved it! Thanks very much, Colin!

I obtained a copy of the Hubley Toys 1972 Catalog, which features a series of four puzzles designed by Piet Hein. With the acquisition of the Twitchit, I now have an example of each of these vintage Piet Hein puzzles.

My Solution to Drive Ya Nuts

The following is my graphical (or tabular) solution to the Drive Ya Nuts puzzle. I have not seen any solution technique like mine applied to this type of puzzle - even Jaap's page says one must try all combinations. My technique is a considerable savings and allows a solution - and negative results - to be derived easily by inspection - the hallmark of a graphical technique. I have worked this entirely "by hand."

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.


I begin by developing a "Primary Table." The table contains one row for each nut A through G, and one column for each number 1 through 6. Each cell contains the 3 consecutive numbers, in a COUNTERCLOCKWISE direction around the nut, that appear on the nut determined by the row, when the number indicated by the column abuts the central nut.

When a central nut is chosen, the remaining nuts must be arranged around it. For any side of the central nut, the orientation of each remaining nut that can abut this side is given by the cell in the apropos row and column of the primary table. The row corresponding to the central nut is eliminated from consideration (indicated by the green line through it). If we re-arrange the columns to correspond with the CLOCKWISE side numbering of the central nut (and repeat the "1" column last, for convenience in analysis), we arrive at the 7 diagrams shown below.

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 all easier than it sounds :-) You go about crossing off cells until you arrive at a solution or an impossible situation. In the diagrams below, I have crossed out cells in red and given lower-case letters to the slashes to indicate the order of my logic. I have circled each impossible situation in purple.

When A is in the center there is no solution possible. Consider nut G. Its 1 cannot be used to abut the central 1 on A, since there can be no nut clockwise from it that matches its 6 while also matching the central 6 required at that position. Hence cell G1 crossed out with line a. G cannot be used to abut the central 6 since there can be no nut counterclockwise from it that matches its 1 while also matching the central 1 required at that position. Hence cell G6 crossed out with line b. Similar arguments apply resulting in the cell in every column of row G being crossed out. This means that G cannot be used, violating rule (1) and proving that nut A cannot be used in the center.

On to nut B in the center. No nut fits counterclockwise of C5, D5, E5 or G5. No nut fits clockwise of A5, or G2. This leaves only nut F possible to abut the central 5, but nothing remains to fit clockwise of it (only another copy of the F nut would fit). This proves that nut B cannot be in the center.

Here, G6 is eliminated - nothing fits counterclockwise of it. This in turn eliminates D4 and G4. Nothing fits clockwise of A4 or E4, or counterclockwise of F4. This leaves only nut B possible to abut the central 4, but nothing remains to fit counterclockwise of it (only another copy of the B nut would fit). This proves that nut C cannot be in the center.

If you logically eliminate all impossible cells from the table when D is in the center, you find a single solution indicated by the six cells circled in blue. This is the only solution to the Drive Ya Nuts puzzle.

When E is in the center, only A can abut the central 1. This then requires C4, but no nut fits counterclockwise from it.

Nothing fits clockwise of G1 (except G again). Nothing fits clockwise of A4, B4 or G4. Since G1 is eliminated, now nothing fits clockwise of D4 or E4. This leaves only C4 but nothing fits counterclockwise of it.

Lastly we tackle G in the center. Nothing fits counterclockwise of B2, A3, C3, D3, or E3. Since B2 is eliminated, nothing fits counterclockwise of F3 either. This leaves only B3 but nothing fits clockwise of it.

Route-Building

Route Building puzzles are a sub-class of Edge Matching puzzles ("Continuous Path" or CP Edge Matching puzzles according to Jacques Haubrich's classification scheme). Here, one has to arrange pieces so that connections are made, creating a specific route across the pieces according to some rule.

 
La Chaine Sans Fin - The Endless Chain
A vintage version, and a recent laser-cut wooden version by Steve Kelsey.
Arrange the pieces in the tray to form a closed loop of chain links.
JH Vol.2 p244

Tantrix is a well-known modern route-building puzzle/game. [Jaap's Tantrix page.]

Psyche-Paths
(c) 1969 Funtastic, Division of KMS Industries, Inc., Alexandria, VA.
A set of 85 hexagonal tiles, each with 3 path segments using up to 3 colors.
A solitaire challenge is to build a 9x10 rectangle alternating rows of 9 and 8 tiles.
Listed at boardgamegeek.com.
Tiles shown here.
Play a related game, Kaliko, online here.

Here is the included solution to another solitaire challenge:


Rubik's Tangle set of #1 thru #4, and the 9 double-sided plastic tiles version
The original sets have 25 one-sided cardboard tiles, and come in four versions distinguished by which particular tile is duplicated within the set.
JH Vol.2 pp272-3

Chicago L - George Miller

I believe this is the "IQ Chain" puzzle, by D.P.B. Taiwan 1993
JH Vol.2 p172 - 1 soln for a single continuous path.

Krazee Links - Plastrix 1939
JH Vol.2 p207

The Path - from Family Games

It's Knot Easy - Milton Bradley
JH Vol.2 p240

Trail Run

Think Through - Pressman
(Equivalent to QED's Set Squares which I don't have.)

Right Connections - Springbok
JH Vol.2 p179

Fishy - Melissa & Doug

Snake Pit - Nature's Spaces / Binary Arts

Octopus - Nature's Spaces / Binary Arts
JH Vol.2 p184

Go-Getter 3 Prince and Dragon, and Cat and Mouse - DaMert
Designed by Raf Peeters - see his website at www.atelier-oker.com

The Path - Waddington's
JH Vol.2 p197
(See the Waddington's series, below.)

Ant Trails

The Diamond Dilemma puzzle was Copyright by Price Stern Sloan Limited in 1989, and offered prizes for solutions of various complexity. The instructions tell you to "arrange the playing pieces on the diamond so that a continuous unbroken line is formed."

This is Dice Dominoes. The box says "Made in U.K." but there's no other provenance. There are twelve cubes, each side of which shows a correctly linked arrangement of two or three dominoes. Paraphrasing from the instructions: Using the box base as the playing area, start with a double-six in the top left corner. (There are six faces among four of the cubes showing a double-six.) The box holds a rectangle of 3x4 cubes. Match dice so a continuous pattern is formed, as in regular dominoes. You must use all 28 dominoes and cannot use any domino more than once. Each must line up and doubles must be at right angles. A solution sheet (I haven't looked) is enclosed.

TSL Maze
Assemble the pieces to form a maze.

Daily Sketch Jig-Saw Puzzle
Fit the pieces into the tray and build a loop.

Serpentiles - Thinkfun

Metroville from Smart Games, designed by Raf Peeters - see his website at www.atelier-oker.com

HooDoo Loop by Popular Playthings
Build a path using the 14 pieces.
Designed by Bill Hanlon and Steve Wagner
A gift from the folks at Oy Sloyd Ab. Thanks!

Cobra Cubes, from SmartZone Games.
Designed by Ariel Laden.
Purchased at Eureka.

Four cubes, each a different color. Each cube's sides have various segments of a snake - a head, tail, or body section. A booklet of challenges, graded A, B, C, or D, requires one to arrange the cubes in order to form a snake that spans all visible faces of the required cubes.


Anaconda - designed by Raf Peeters
A 5x5 grid with one corner filled, seven 2-sided pieces, and a 1x1 blank piece. Cover a designated square with the blank, then fill in the rest with the other pieces while building a complete snake.

Tromino Trails from Pavel Curtis
IPP31 Berlin
Using subsets of Tromino pieces inscribed with trail segments, make a single closed loop in five different-sized openings in the adjustable tray.

String Octet
Designed by Goh Pit Khiam, made by Walter Hoppe. Purchased from Walter at IPP 29 in SF. 8 1x2 tiles encompassing all the ways a tab and a notch can be arranged, each with a path inscribed connecting the tab and notch. Two 1x1 tiles, one with a notch and one with a tab - the start and end tiles respectively. A 4x5 tray. A sheet with 13 challenges - each specifies positions and orientations for the start and end tiles. The objective is to connect the 1x2 tiles in the tray to create an unbroken path from start to end. I really like this one!

World Passport, from SmartZone Games.
Designed by Tzafrir Kazula.
Purchased at Eureka.

Fit six poly-hex pieces into a hexagonal grid overlaid on a "country" card such that five points are connected in order by a transparent path, with no backtracking.


Troy Extra Muros from Smart Games, designed by Raf Peeters. I really like this one!
A series of challenge cards indicate the placement of up to 4 blue and 4 red soldiers. Place the four wall pieces such that all blue soldiers are completely enclosed and all red soldiers are "outside" the walls ("Extra Muros").
(You are finding the route the walls must take.)
Raf gave an interesting talk at IPP31 in Berlin, in which he said that one goal of design is to ensure a customer can apprehend the goal of a puzzle just by looking at it in the package. He achieved that in this case - I could tell what the goal was despite the fact that I found this in Québec City and this box was printed in French!

Golf Line by Binary Arts 1994
Arrange 20 1x1x2 rectangular cards depicting sections of an 18-hole golf course to form a continuous cart path from hole to hole.
Hints and solution are included.

Mall Function, designed and exchanged at IPP32 by Rik van Grol, made by Buttonius

Network Builder, designed, made, and exchanged at IPP32 by Wei-Hwa Huang

Temple Trap - designed by Raf Peeters, issued by Smart Games
A 3x3 board with an exit space in one corner. Eight sliding blocks representing walls, hallways, and stairs in various configurations, and a player figure. Set up the blocks and figure per the current challenge, then try to solve it by sliding the blocks and moving the figure from block to block to eventually exit the board. The block where the figure resides cannot be moved, and the figure can only be moved to rest on certain blocks. Temple Trap combines a sliding-piece puzzle with route-building challenges and has a great theme reminiscent of Indiana Jones.

Perplexing Python issued by Pentangle

Roundabout - invented by Volker Latussek
Produced by Popular Playthings
Use the 18 pieces of four different shapes to build a closed loop - assemble 50 challenges.

City Maze - designed by Raf Peeters, produced by SmartGames
Thanks, Raf!

 

Silhouette Puzzles

This category requires you to arrange the pieces to satisfy some rule or goal relating to a pattern/silhouette the pieces make. There is no physical mechanism to restrict moves - only rules or the goal govern legal combinations. The piece shapes will be fairly abstract but usually it will be easy to abut them and they will interchange positions easily.

Note: I have created a separate page for Tangrams.


This classic three-piece French puzzle is called Bucephale. Arrange the three pieces to form a horse. Described in Slocum and Botermans' "New Book of Puzzles" on page 23. Sam Loyd called it his Pony Puzzle.

Demon Dino - William Waite

Dino Try - William Waite

Six Australian Animals designed by René Dawir

Cliko, by Foxmind
Comes with a set of blocks and a booklet of problems. Arrange specific blocks to form different silhouettes.
With the Tangramino, Equilibrio, and Architecto books from Foxmind, I can use the pieces in the Cliko set I already have, to try many new challenges.

Matchstick Puzzles

Using just a set of matchsticks (or sticks without the matchheads), form a figure, then transform the figure into some other figure moving only a specific number of matchsticks.


Puzzle Picks, by Kohner (No. 122) 1967. Includes a set of colorful plastic "matchsticks," a booklet of 80 puzzles, and a solution sheet.
There are lots of matchstick puzzles on-line...

The book Creative Puzzles of the World by van Delft and Botermans includes a section on matchstick puzzles on pages 49 through 56.


A set of brightly colored matchsticks, in a matchbox marked "Puzzle" from Japan.

Allumez les Enigmes, produced by Kikigagne. A set of matches and cards posing 50 matchstick challenges (in French). Thanks, Brett!

33 Problems - issued by Invicta in 1981
A board containing a set of grooves, arranged so as to provide a playing surface on which to try 33 different matchstick puzzles using a bunch of included red panels in lieu of matchsticks. The panels fit into grooves in the board. An included sheet specifies the setup and objective for 33 puzzles, but does not give solutions.

Stacking / Overlay / Overlapping Pattern Puzzles

In this category of puzzle, a set of pieces, often transparent or with cutouts, must be stacked (or folded) over one another in order to achieve some goal.


The Rabbit Silhouette or "Question du Lapin" - layer the cutouts to form a rabbit (play the rabbit silhouette on-line).

Arrange the 5 layers so the cumulative cutout area forms the shape of a pipe. You will then have "Not A Pipe" - "Ceci n'est pas une pipe" - as from Rene Magritte's "The Treason of Images." A miniature puzzle in a matchbox.

Mattel's Virtual Illusion puzzle contains a base and a series of transparencies each containing a portion of a three-dimensional image. You need to order the transparencies in the base so that the image appears correctly.

Two Silhouette Puzzles designed by Diniar Namdarian- make a Pigeon, and make a Dog.


The Cryptic Classics series from the 1990's was issued by Crystal Lines and sold by various parties including Binary Arts, and Buffalo Games Inc. (BGI) of Buffalo NY. Each is a modern adaptation based on an old puzzle design. The "Create a Panda" puzzle is based on an old puzzle called "Milk" described in Slocum and Botermans' 1992 "New Book of Puzzles" on page 15. The Cryptic Classics series includes a 3-part "Seat the Riders" puzzle, and "Find the Escapee" which is similar to the classic vanish "Get Off the Earth."

Toyo Glass issued a series of puzzles where you stack clear glass coasters with various patterns:

Red All Through

Animal Land

Starry Skies

Along the same lines as the Toyo Glass puzzles, combined with the weave concept, Strip Tease requires you to create a 5x5 weave using 10 clear strips having various arrangements of quarter-squares so that all solid squares result.

Trixxy designed by Dror Green - superpose four cards having transparent and opaque colored sections in order to produce a solid column of each of the four colors.

This is the "2D 3D Burr" - stack the transparencies so that the image of a 3-piece burr appears.
I bought it at Torito.
The Transposer series of puzzles has been created and developed by Albatross Games Ltd. of London, and distributed by the Toysmith Group. Available from In and Out Gifts. Each puzzle consists of a set of cards with various design fragments and cutouts. Stack the cards to achieve specific patterns, such as uniform color front and back, or unbroken paths of given colors from point to point.


Transposer 6

Transposer Bonbons

Transposer Genesis

Transposer Kaboozle

Transposer Tiffany

Transposer Tower of London

Transposer Struzzle

Cover Your Tracks - Thinkfun
Four pieces and a set of challenge cards - for each card, pack the four pieces into the tray so that the bootprints on the card are all covered.

Zoo Panic
by Tsugumitsu Noji
Purchased at IPP28 in Prague.
Four transparent overlays, each with a 6-unit enclosure. On each problem background, fence off unlike things.

Frustables
by Gameophiles Unlimited 1973
Six sets of six cards. Each set of six cards has one colored puzzle on the front and another color on the back - twelve puzzles in all. A card has a given color, and may have some black area. The objective given the six cards belonging to a given puzzle is to pile the cards such that all the black areas are covered but no colored area is covered. These are difficult puzzles!

Ariel Laden's Kookoo Puzzles - Funny Fliers
Four six-card puzzles, and one large puzzle using the backs of all 24 cards.
Position and interleave the cards of a set to form a complete picture.
This principle is very similar to that of the 1973 puzzle Frustables by Gameophiles Unlimited.

Four well done puzzle challenges from Smart Games.
North Pole Camouflage, Safari Hide & Seek, Traffic Control Airport, Pirates Hide & Seek
Booster packs for North Pole Camouflage and Safari Hide & Seek
Designed by Raf Peeters - see his website at www.atelier-oker.com

Angry Birds Playground Under Construction - designed by Raf Peeters, issued by Smart Games
Using the four pieces (non-flippable F, P, T, and U pentominoes), cover the proper selection of faces on the 5x5 gameboard, per each challenge. I admit I had trouble with the very first problem!

Mozaniac: Four Painters
By Paradoxy Products.
Six square cards, each almost cut into quarters and each bearing a portion of four different images. Interlock and overlap the cards to create each image, in turn.

The Invisible Puzzle, designed by Rich Garner, from Loncraine Broxton - Lagoon Trading Co. Ltd. Make a large hexagon from 18 transparent trapezoidal tiles, while matching edge colors.

On the Dot - Gamewright

Get a Clue from Pavel Curtis.
A refined version of Pavel's exchange puzzle from IPP30.
Nine transparent pieces, and a frame in the shape of a magnifying glass.
Arrange the pieces in three layers inside the frame.
Find two distinct overall arrangements to determine "two things given to Snow White."
I enjoyed this one - it is not too difficult.

Pathwords
invented by Derrick Niederman
For each of 40 challenges of graduated difficulty, fit a subset of the supplied transparent pieces to completely cover the field of letters such that each piece covers one word that runs backwards or forwards.

Cage the Animal, designed by Manabu Satou and Naoyuki Iwase (Osho), made by Woodpecker Hikimi and Lixy, exchanged at IPP32 by Lixy Yamada

Daily Puzzle - designed by Wei-Hwa Huang and Oskar van Deventer, produced by Thinkfun
Six tiles must be paired and overlaid in variuous ways to form 3-latter month abbreviations, and four tiles must be similarly arranged to produce the date. Each tile has either black segments, cutout segments (through which whatever is behind the tile, including the black base) will show, or blank area.

Smart Tease - from Kidult
14 strips, each having seven colored dots on one side
and on the other small pins and sockets whereby strips can be joined.
Each dot is one of seven colors.
Form a 7x7 mat by joining strips back-to-back cross-wise, such that
where two strips cross the dots on both sides are the same color.
   
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.

The Eight Queens and Other Positioning Puzzles or Configuration Problems

The Eight Queens Problem requires you to place 8 queens on a standard 8x8 chessboard such that none attack each other - i.e. no two queens can be in the same row, column, or any diagonal.

It is discussed by Slocum and Botermans in their Puzzles Old and New on p.37.

This problem first appeared in the 1848 Berliner Schachzeitung, where only two solutions were provided (c.f. Singmaster Source2.doc), and was proposed by Bavarian chess master Max Friedrich Wilhelm Bezzel (pictured at left). Bezzel was born in 1824 in Herrnberchtheim, one of six brothers and three sisters; he was a math teacher and lawyer, and died young at age 47 in 1871, probably of cancer.

The full solution to the problem did not appear until published in the Leipziger Illustrierte Zeitung 15 No. 377 of 21 Sep 1850 by mathematician Franz Nauck (1815-1902).

A proof that there are only 12 unique solutions was published in the 1874 Philosophical Magazine by English mathematician and renowned pottery collector James Whitbread Lee Glaisher (1848-1928), pictured at right. The text of PM is available online.

Each solution has 8 rotations and reflections except if it is 180 degree symmetric in which case it has only 4 - only one of the 12 is 180 degree symmetric, so there are a total of 11*8 + 1*4 = 92 distinct solutions including rotations and reflections of the basic unique 12.

The problem is generalizable to the N Queens Problem, using n queens on an nxn chessboard. There is no known formula for computing the number of solutions given n. You can see a chart of findings for various n (up to 26) at www.durangobill.com/N_Queens.html . Solution counts for the first few n are given below...

N 1 2 3 4 5 6 7 8 9 10 11 12
Unique 1 0 0 1 2 1 6 12 46 92 341 1787
Total 1 0 0 2 10 4 40 92 352 724 2680 14200
The problem is discussed by Edouard Lucas (1842-1891), inventor of the Towers of Hanoi puzzle, in his 1895 book L'Arithmetique Amusante (available online), in which he gives the nice table of the 12 basic solutions shown below.

Over the years there have been many instantiations of this problem posed as mechanical puzzles.

 
The Brain Drain - Reiss 1979
and Save the Queens - a recent remake by Dave at Puzzle Crafthouse.

Arrange the four squares such that in the resulting 8x8 grid, no two holes appear in the same row, column, or diagonal. The tiles may be flipped over and rotated.

I have had this puzzle for a long time and it remains one of my favorites despite its simplicity.

If you examine the 12 unique solutions to the 8 Queens puzzle and divide each into quadrants, you'll find that there are only six types of quadrants that each contain two queens. In the diagram at right I have labeled each quadrant type, regardless of rotation or reflection, assigning a letter A through F. Two of the solutions, labeled VI and VIII by Lucas, have a single queen in two quadrants and three queens in the other two - I haven't labeled those quadrant types since they cannot be used in the Brain Drain puzzle.

This table summarizes the use of the six quadrants in each of the relevant 10 solutions. Only one set - those from solution V, has no duplicate of a quadrant piece type, and has a unique solution. This is the Brain Drain set!

Solution
ID
Quadrant
Types
Used
Comments
IAABB Dup. Pcs.
Mult. Solns.
IICCDDDup. Pcs.
IIIAABBDup. Pcs.
Mult. Solns.
IVABDEMult. Solns.
See X.
VACDF No dups;
One Soln.
VIn/a# Q/q != 2
VIIAAFFDup. Pcs.
Mult. Solns.
VIIIn/a# Q/q != 2
IXAAFFDup. Pcs.
Mult. Solns.
XABDEMult. Solns.
See IV.
XIADDFDup. Pcs.
XIIAAFFDup. Pcs.
Mult. Solns.
 
The Frustr8tor
From my friends at Puzzlemaster.ca - thanks, guys!
28 versions of the 8 Queens to solve. The front side shows an 8x8 grid. The back side has 8 tracks corresponding to the 8 columns of the grid. Along each track, each of the 8 row positions is marked by a number from 1 to 28 - some appear three times, some two. One red and one green tab ride in each track - green at the top and red below. To try a puzzle, choose a number and set a red tab at every position marked by that number. Then, using the green tabs in the remaining six columns, fill in the grid according to the usual rules - a dot appears in the grid on the front at the position where a tab is set.
 
This vintage French boxed version called "Jeu des Manifestants" uses 8 tiles, and is available in two versions with either triangular or rectangular tiles. Jeu des Manifestants is shown in Slocum and Botermans' "Puzzles Old & New" on page 37.

For another version using battleships, see U.S. Patent 1151615 - Reibstein 1915.


Lots O Spots by Peterson - "an L. J. Gordon creation."
16 tiles, each divided into four quadrants, and with 1 or 2 spots on each. When the tile are arranged in a 4x4 grid, the quadrants define 8 rows and 8 columns. The spots come in three colors - red, purple, and green - and are distributed so that there are 8 of each color. In the hardest of four challenges, you must arrange the tiles in a 4x4 grid such that all rows, columns, main diagonals, and all short diagonals contain no more than one spot of each color. This amounts to solving the 8 queens problem simultaneously for 3 colors of queens.

The Schpotz puzzle by Peterson Games. Arrange the nine tiles in a 3x3 grid such that every row, column, and main diagonal contains exactly three spots.

This is Out of Line by Crisloid Plastics of Providence RI.
The objective is to arrange the eight pegs in the 8x8 grid so that no more than one peg is in any row, column, or diagonal. In addition, at least one peg must be in each of the five differently shaded areas - one of the areas is a single position at the lower right corner. The cover shows a motorcycle gang - I think they're supposed to be "out of line."

Hoo-Doo from Tryne, circa 1955.

An 8x8 board, with inner nested 4x4 and 6x6 areas marked off. 64 pegs, 8 each of 8 different colors, and two extra "neutral" pegs. There seem to have been at least two different sets of instructions issued with this puzzle.

 

The first two challenges cited on the first instructions, on the 4x4 and 6x6 grids, are easy since you can use more than 4 and 6 colors respectively. However, the 8x8 challenge amounts to superimposing 8 solutions of the 8 queens problem, and is impossible! This puzzle is interesting because it requires superposition, and also because it seems to be one of those rare instances where prize money is offered for an impossible challenge.

My logic is based on an argument by Tom Jolly I read over at the PuzzleWorld forums. Think about the two main diagonals - all 16 spaces must be filled, but some reflection should convince you that each of the 8 colors must contribute exactly two pegs somewhere on the main diagonals - one on each. If a solution failed to contribute, that would mean two spaces on one diagonal would have to be filled by another color - which would violate the rules. That means that the only usable solutions from the 12 are the six that have a queen on each of the main diagonals.

But, each of those six solutions also has a queen on one of the penultimate corner squares.

Since every possible rotation and reflection of any of those six solutions will also have a queen on a penultimate corner square, and there are only four such squares to go around, we cannot superimpose more than four solutions before we have a conflict!

I did more research into this puzzle and found that the superposition problem has been discussed by Martin Gardner in his book The Unexpected Hanging and Other Mathematical Diversions. Martin writes that "When the order of the board is not divisible by 2 or 3, it is possible to superimpose n solutions that completely fill all the cells. Thus on the 5x5 one can place 25 queens - 5 each of 5 colors - such that no queen attacks another of the same color." Unfortunately, Martin gives no proof or attribution. In the Chapter 16 addendum, however, a reader points out a reference to a proof of the impossibility of the 8 color superposition, given by Thorold Gosset in the Messenger of Mathematics Vol. 44, July 1914, on page 48. (The MoM is available online.)

Gosset's proof and mine are similar, but I think mine is more elegant :-)

A solution for 6 superpositions on the 8x8 is given by Lucas in his 1895 book. Of note is that only four of the superimposed six require a penultimate square - the other two do not - but not all main diagonals are filled.

Now, can you find the superposition for 5 colors on a 5x5? What about for n=7 or n=11?

This puzzle is called Orchard and is offered by the Australian company Dr. Wood in their Mind Challenge series. I don't have a copy, but it poses another interesting superposition problem.

You are given a board having an 8x8 grid of sockets, with a 2x2 house obstruction along the center of one side. You are also given 40 trees, 10 of each of 4 types. You are to plant the trees in the grid such that each set of 10 forms 5 rows with 4 in each row.

This type of problem is discussed by Prof. David Singmaster in his Sources in Recreational Mathematics in section 6.AO, Configuration Problems. Singmaster defines a notation to describe the various related flavors of this type of problem: (a,b,c) meaning arrange a points in b rows of c each. Singmaster does not point out the earliest appearance of this puzzle, but cites many examples in the puzzle literature.

The Orchard puzzle calls for a superposition on an 8x8 grid, with the obstruction, of 4x (10,5,4). Singmaster cites Dudeney's 1917 book Amusements in Mathematics (available online) for an example of a (10,5,4) problem, and notes that Dudeney describes six basic solutions.

Dudeney states that there are only six fundamental solutions to the problem of arranging 10 points in 5 rows of 4 each (though each of the six patterns can be infinitely distorted depending on the overall underlying grid size), and he shows diagrams with names.

Dudeney also gives the minimum grid size required to form each of the patterns: Star=7x9, Dart=7x7, Compasses=12x17, Funnel=8x8, Scissors=9x11, and Nail=7x11. Only the Dart or the Funnel could be used on the 8x8 grid of the Orchard puzzle.


Funnel

Dart (a)

Dart (b)

Dart (c)

The Funnel has only four rotations/reflections/translations within the 8x8 grid. When four copies are superimposed, it turns out the points conflict with the house obstruction, so the Funnel shape cannot be used in the solution to the Orchard puzzle.

The Dart shape can be stretched and moved around in the three configurations shown. The (a) and (b) versions also ultimately cause a conflict with the house. However, four copies of the (c) version, suitably rotated/reflected/translated, can be superimposed so that they do not conflict with the house. This is the solution to the Dr. Wood Orchard puzzle!


The Pin and Dot puzzle - "Insert six pins, each in a separate dot, so that no two pins shall be on the same line." This is equivalent to a six-queens puzzle.
(I don't have this - shown for reference.)
The six-queens puzzle has only one unique solution. Since that solution is 180-degree symmetric, it has only four rotations and reflections.




 

 
La Strategie des Nippons, Strategie des Boers, and Nouvelle Strategie
These vintage French boxed puzzles are all instances of the six-queens puzzle.
(I don't have these - shown for reference.)

Jeu des Sentinelles
"A police chief represented by the red piece, with 7 sentinels represented by the white pieces, must position himself and his sentinels so that no man can see any of the other men along any straight vertical, horizontal, or diagonal line."
This vintage French boxed puzzle is an instance of the eight-queens puzzle.
(I don't have this - shown for reference.)

L'Intraitable - "The Intractable"
Given a 6x6 grid and 24 tokens - six each of four colors, arrange the tokens on the grid so that no horizontal, vertical, or diagonal line contains more than one token of the same color.
This vintage French boxed puzzle is an instance of a superposition of four six-queens puzzles.
(I don't have this - shown for reference.)

The four rotations and reflections of the six-queens solution can be superimposed:

 

L'Embarras du Brigadier and L'Embarras du Caporal
Given a 5x7 grid of points and 12 men, arrange the men on the grid to form six rows with four men in each row. Both of these vintage French boxed puzzles are instances of, using Singmaster's notation, a (12,6,4) configuration problem.
(I don't have these - shown for reference.)

Singmaster cites several appearances of the (12,6,4) problem, including in The Sociable of 1858 (as Problem 26, The Gardener's Puzzle), The Book of 500 Curious Puzzles of 1859, and Hoffmann (1893) Chapter VI, No. V.

There is a second solution (see The Sociable, or Hoffmann), but it does not fit on the 5x7 grid.

Note that some (a,b,c) configuration solutions rely on the trick of stacking more than one token at a given point.

The Five-Queens puzzle has only two unique solutions.

 

The asymmetric solution has eight rotations and reflections. The symmetric solution has only two, giving a total of 10. A selection of five can be superimposed.

The Seven-Queens puzzle has six unique solutions, and 40 rotations/reflections.

   
   

Combination Puzzles


SafeCracker 50 - a reproduction of a classic design style, from Dave Janelle at Creative CraftHouse. Thanks very much, Dave!
Arrange the five layers so that the five numbers appearing in each of the 16 radial columns total 50. There are many vintage paper/cardboard puzzles using this theme - Dave's is made from laser-cut wood and is far sturdier. With the base plate and four additional movable disks, there are 65,536 combinations to try!

This "SafeCracker" type of puzzle was the subject of U.S. patent 993581,
granted to H. Davidson in 1911.
But, see below for versions with earlier dates...
The Davidson patent seems to have been applied to the "Great Burglar Puzzle" which requires columns to total 40. You can see the reference to the patent near the center. (I don't have this.)

Even earlier, the Hegger's Century Puzzle says "Patent applied for, Copyright 1891 by G. A. Bobrick, New York." (I don't have this puzzle.) G. might stand for Gabriel, since he holds other patents from the late 19th century. Unfortunately I could not find a patent pertaining to the puzzle. While the USPTO website does allow searching for applications (as opposed to granted patents), the database only contains records from 2001 onwards. The USPTO allows one to search for granted patents from 1790 through 1975 but only by issue date, patent number, and current U.S. classification.

For reference, here are several more vintage "SafeCracker"-type puzzles (I don't have these):

The White Sewing Machine Sixty Puzzle says "Copyrighted by A. L. Vandyke 1890." There doesn't seem to be a date on "The Great '51' Puzzle."

Note the swastikas that appear on the Huntley & Palmers puzzle - the symbol actually means "that which is associated with well-being" so this likely predates WWII and the darker usage of the symbol and was probably issued somewhere in the period from 1841 (when Palmers joined Huntley) to 1920 (when the Nazis rose to power). The "Capture U-Boat 40" puzzle, however, obviously comes from the WWII period. It was manufactured by the Geiger Bros., Newark N.J.


The following photos are from Jerry Slocum's collection, courtesy, The Lilly Library, Indiana University, Bloomington, Indiana. They can be found under classification section 1.3, Miscellaneous Put-Together.

"The Jayne Problem" is Copyright 1906 by George H. Vickers. I don't know what is on the back of the Leinbach puzzle (another sum-to-40 type) or the "Safe Combination Puzzle" (make all 12 columns sum to 55), but their fronts display no dates.

The Hecker's puzzle says "Pat. applied for" and "Since 1840" so probably dates post 1840, but displays no other info and I don't know what the back might say. The Steinfeld puzzle says "Pat. June 20th, 1882" - with a bit of research I found that the patent in question is US259920 issued to William H. Reiff of Philadelphia on June 20, 1882. That makes this the oldest version as far as I know.

Dave at Creative Crafthouse has brought back the 1882 puzzle and issued the Word Wheel. Thanks for the copy, Dave!

Latin, Graeco-Latin, and Magic Squares

Latin Squares

According to the Wikipedia entry, a Latin Square is an NxN matrix filled with N different symbols such that no symbol appears more than once in any row or column. Sometimes the additional restriction of disallowing a repeated symbol along either main diagonal is also added. See Terry Ritter's page, Latin Squares: A Literature Survey for a nice collection of facts and terminology about Latin Squares. Also see a nice article by Elaine Young.

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, Rn, the number of distinct squares Ln is:

Ln = n!(n-1)!Rn

Order
n
Enumerator(s)
Year
# Reduced
Rn
# Distinct
Ln
2Euler
1782
12
3Euler
1782
112
4Euler
1782
4576
5Euler
1782
56161,280
6Fisher and Yates
1934
9408 
7Norton
1939
16,942,080 
8Wells
1967
535,281,401,856 
9Bammel and Rothstein
1975
377,597,570,964,258,816 
10McKay and Rogoyski
1995
7,580,721,483,160,132,811,489,280 

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 Safari Puzzle
by the German company "Pussy" or "Pussycat."
A 7x7 Latin Square puzzle presented as a sliding-piece puzzle. Arrange the pieces (7 each of a Hippo, Lion, Elephant, Rhino, Giraffe, Zebra, and Tree) so that only one of each appears in each row and column.

More Madness
by Parker - The Fun and Game Name. No Date.
It comes as a single 5x5 plastic sheet, scored with grooves along which you are to break apart the pieces. Each square is one of five colors - yellow, red, blue, green, or white. The unbroken sheet shows the solution - this is a Latin Square puzzle and in the grid, no color occurs more than once in each row or column. There are nine pieces - two are 1x2, seven are 1x3.

Bird's Puzzle, by Chad Valley.
The Bird's Puzzle is very similar to the More Madness puzzle. There are nine pieces, two 1x2 and seven 1x3, colored with five colors - yellow, red, blue, green, and a Bird's logo - to be arranged into a 5x5 grid such that no color appears more than once in each row and column.

The Missionary Puzzle and Four Others - Five (5) Old Time Puzzles
by The Embossing Company of Albany NY.
Includes: Missionaries and Cannibals; Staggered Colors; Change About; Double Up; and Sorting Out.

A vintage French boxed puzzle called Les 15. An order-5 Latin Square.

Testa Cross Colour
Same as Bird's.

A vintage cardboard advertising puzzle, "Say Cheese Louder." Nine pieces to be arranged in a square such that only one of the five symbols appears in each horizontal or vertical line.
I have highlighted the scored lines separating the pieces.


The "Sudoku 9 Abstract" puzzle from the Plantecs at PuzzleMePlease.com.
Form a 9x9 Latin Square using the pieces made from colored woods.

Utopia issued by Popular Playthings 2009
invented by Sjaak Griffioen
Sixteen "buildings" - four each of four different heights.
Place on the 4x4 grid according to rules and hints given
on 50 challenge cards divided into 25 Phase 1 and 25 Phase 2.
I enjoyed Phase 1 but Phase 2 seems overly confusing.
(NOTE: this may seem to belong in the Graeco-Latin section, but notice that all buildings of the same height are the same color, so height is really the only differentiator.)

Graeco-Latin Squares

A Graeco-Latin (or Greco-Latin) Square (also known as an Euler Square) is constructed by superimposing two Latin Squares having the same order but different sets of symbols (usually designated by using Latin letters for one of the squares' symbols and Greek letters for the other, hence the name Greco-Latin), such that each combination of symbols (one from each Latin square) occurs only once in the superposition. They are also known as mutually orthogonal Latin squares or MOLS. There are none of order N=2, but N=3,4, and 5 all exist. While searching for (and failing to find) a solution to the Thirty-six Officers Problem, Leonard Euler conjectured that solutions don't exist for any order N = 4i+2 (i.e. 2, 6, 10, 14, etc.). Euler demonstrated methods for constructing Graeco-Latin Squares when N is odd or a multiple of 4.

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:



Bali Buttons
Requires you to place the sixteen tokens - all combinations of four different shapes with four different colors - on the 4x4 board so that no row, column, or main diagonal contains more than one token with a given shape or color. This is a Graeco-Latin Square puzzle.

Four Square
from the Embossing Company, Albany, NY
Combines a Graeco-Latin Square puzzle with two sliding piece challenges.

FIRST. Place the blocks in the box so that no two of the same number nor of the same color are in any of the 10 horizontal, vertical, or diagonal lines.

SECOND. Remove one of the 4's, then, by sliding them about, arrange them in horizontal rows, each of a different color, and in the order of 1,2,3,4. The fourth or bottom row should be 1,2,3.

THIRD. After completing the second, slide them about again to arrange them as in second, but in a vertical position.


Brain Strain
An advertising puzzle consists of sixteen small playing cards - the Jack, Queen, King, and Ace in each suit. As with any Graeco-Latin square puzzle, the objective is to arrange the pieces in a square grid so that neither of the two kinds of feature (in this case, face value and suit) appears more than once in each row, column, or main diagonal. This puzzle was first proposed by Jacques Ozanam.

Quintessence
Copyright 1970 by gametime, Inc. of NY, NY
A 5x5 Graeco-Latin Square - arrange the 25 pieces in a 5x5 grid such that no row, column, or diagonal contains more than one instance of a circle color or tile color.

A Floral Derangement - Peterson Games 1972

A vintage advertising Graeco-Latin Square puzzle issued by Stephens Motor Works "of the Moline Plow Co., Freeport, Illinois." There are 16 cards: four of each color - orange, purple, green, and yellow; four of each car model - roadster (92), touring (94), touring (96), and sedan (93); and each with a "license number" 1 through 16. The objective is to arrange them in a 4x4 grid such that rows, columns, and main diagonals contain cars of all different models and colors, and the sum of license numbers is 34 in all rows and columns and the four corners, the four centers, and each group of four in a corner!

The 36 Cube - Thinkfun
Presented as a 6x6 Graeco-Latin Square - in this case, one feature is the height of the tower, the other is its color. The usual rules would seem to apply. However, remember the proof that no Graeco-Latin Squares exist for order 6?

Some info from Thinkfun's press release: According to Nick Baxter, "Its friendly look fools you into thinking it's easy, but 36 Cube takes perseverance and practical smarts to solve. Its solution lies in an 'a-ha' moment, an insight equally discoverable by almost anyone willing to spend time with it." Created by Dr. Derrick Niederman, an MIT Ph.D. who thought of the idea after running across a 'mathematical supposition from the 18th century' while writing a book on numbers. Hmmm - I wonder which supposition?

You can play online - trying Graeco-Latin squares up to order 5. To try level 6 you've got to buy the puzzle.

Thinkfun also sponsored a contest - I was one of the winners!


The Giant Puzzle
From Dave Janelle of Creative Crafthouse. Thanks, Dave!
Some info from Dave: "The 'Giant Puzzle' was first introduced in 1888 by McLoughlin Bros of New York. We found a picture of the old puzzle in Slocum and Botermans' book, New Book of Puzzles (1992) and were fascinated by it. It has a bit of a Sukoku edge to it, and it's really hard. The object is to arrange all the pieces in the base such that there is no color repeated on any row, column, or any diagonal. Also, no number can be repeated on any row, column or any diagonal. Also, each row, column and major diagonal must add to 25. You might try to tackle each of the requirements separately before you take on the full challenge of making them all happen at once!"

Magic Squares

A Magic Square is to contain a sequence of numbers such that the numbers appearing in each row, column, and main diagonal add up to the same sum, known as the magic constant of the square. If all diagonals (main as well as partial) also sum to the magic constant, the square is a pandiagonal or panmagic square. If replacing each number by its square also results in a magic square, the square is bimagic. If the sequence of numbers used in a square of order n is from 1 to n2, it's known as a normal magic square. Magic Squares exist for all orders except N=2. There is only one distinct Magic Square of order N=3. There are 880 of order 4, and over 275 million of order 5. Supposedly the order 3 magic square was invented in China between 650 and 400B.C. and known as Lo Shu.

4 9 2
3 5 7
8 1 6

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:

Mn = 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.


Fascinating 15 - Crestline
Crestline put out this order-3 magic square -
what could be simpler to manufacture, eh?
(I don't have this.)

Thinking Man's 34 - Skor Mor.
Also 34 Skidoo by Reiss, 1971.
Sixteen wooden tiles printed with the numbers one through sixteen. Arrange them in a 4x4 grid such that every row and column and main diagonal totals 34. Also find an arrangement in each of 12 distinct classes/patterns.

Mystifying 65 - Crestline
An order-5 magic square from Crestline.
(I don't have this.)
     18  17  3
  11   1   7   19
9    6   5   2   16
  14   8   4   12
     15  13  10
There is a single magic hexagon of side 3, with 19 cells. Its magic constant is 38. It was discovered in 1895 by William Radcliffe. In 1964 Charles Trigg published a proof that this is the only magic hexagon of any size (save the trivial single hexagon). Additional historical anecdotes about this puzzle are given in Slocum and Botermans 1994 The Book of Ingenious & Diabolical Puzzles on pages 26-27.

Pinwheel:27 from Are-Jay Game Company of Ohio

 

Domino Puzzles

"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.


Other good resources for Domino Puzzles:

The Instant Insanity Family

One of the earliest mechanical puzzles I ever had was an Instant Insanity given to me by my mother. (Thanks, Mom! BTW, this puzzle obsession is all your fault!)

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).


Here is my copy of the solution letter sent out by Parker Brothers on request:


The Schossow / Katzenjammer Puzzle

The first puzzle in this family was designed and patented in 1900 by Frederick A. Schossow of Detroit. It was marketed as the Katzenjammer Puzzle. An original cardboard container I have is marked "The Katzenjammer Puzzle -- B. W. Gottechalk Patentee Chicago, Ill. U.S. Patent No. 646463 -- Price 10 Cts."

The four blocks were marked with the four suits of a deck of playing cards - hearts, clubs, diamonds, and spades. The Katzenjammer puzzle is described on page 38 of Slocum and Botermans' 1986 book "Puzzles Old & New." According to Slocum, this puzzle appeared in the Johnson Smith Catalog of 1919.

Slocum's book shows the layout of each of the four cubes, as does the patent, pictured here. In the physical puzzle, each block is a different color - orange, pink, green, and yellow. (I have a copy with one block missing, and another complete copy. I am missing the fourth block from the top in the illustration in the book, which I believe is the yellow one.)

"Directions -- Mr. Katzenjammer brought this little box of blocks to his wife, and said to her:- 'Katerina, you will notice that on the top row of these blocks there is a diamond, a heart, a spade, and a club. Now take the blocks out of the box and place them together so that all four sides will have one spot of each kind in a row. It comes easy, Katerina,' he said. 'If you look at the picture on the box, because that has one spot of each kind on two sides already yet.'"

Here is a graph of the Katzenjammer puzzle, side-by-side with the graph for Instant Insanity:

 

It turns out that the Instant Insanity puzzle is isomorphic! I've colored the Schossow nodes to show the correspondence. I numbered the Schossow cubes according to their top-down order in the patent image. I numbered my Instant Insanity cubes arbitrarily - turns out II#1 maps to SK#3 and vice-versa. II#2 = SK#2 and II#4 = SK#4.

I've analyzed over 20 puzzles in this family, and most of them copy Schossow's design. O'Beirne had already remarked that many versions of this puzzle known to him were isomorphic - including the Katzenjammer, Great Tantalizer, and Symington's. The symbols might be replaced or re-mapped, but the essential pattern of the set of cubes is isomorphic.

For example, see patent 02024541 - Silkman 1935. Silkman's cubes are virtually identical - just the symbols have been re-mapped, and two cubes are mirror images of Schossow's, but this makes no difference to the solution.

 

So, how does one establish that two such puzzles are in fact isomorphic?

Well, I read up on the Graph Isomorphism Problem mentioned previously, and found references to nauty by Brendan McKay, which is evidently the best publicly available software. However, it seems like overkill for my purposes.

One also needs a way to represent graphs as data - there seem to be two good standards, DOT and GXL. I've chosen DOT since the graphviz package is handy.

I categorize these puzzles by a numeric shorthand: the number of faces per element, the number of elements, and the number of colors/symbols in total. II-type (Instant Insanity - type) puzzles, with 6 faces per cubic element, four cubes, and four colors, are (6,4,4) in my scheme. (6,4,4) puzzles are represented by pseudographs having exactly four nodes and exactly twelve labeled, undirected edges. For now I'll omit puzzles with fewer or more cubes, non-cubes, and fewer or more symbols/colors.

Many approaches to graph isomorphism entail establishing some way to convert a graph to a canonical form, and showing that two graphs with the same canonical form are in fact isomorphic.

Another useful concept is the invariant - some property of a graph, that all graphs isomorphic to it must share - e.g. the number of nodes. Obviously, two invariants that apply here are four nodes and twelve edges. One can also try to define a complete or sufficient invariant which completely decides isomorphism - the number of nodes and edges doesn't meet this need. Yet another useful concept is the signature or certificate of a graph - a way of assigning a coded ID to the graph as a whole, that plays the role of a sufficient invariant.

I've come up with what I believe is a good way of assigning certificates to II-type puzzles.

First, encode the nodes using four hex digits (0-F) based on their connectedness: The rightmost or least significant place counts the number of self-loops on the node. The other three places give the counts of connections to the other three nodes in no particular order, but sorted in increasing value from left to right. For twelve-edge graphs, we really only need up to digit 'C' and in practice most counts are only as high as 3.

For the II/SK puzzle, the four node codes are: 0221 (the green or heart node), 0230 (the blue or diamond node), 1221 (the red or club node), and 1230 (the white or spade node). This is convenient since each node gets a unique code.

Next, encode the cubes themselves, each as a series of six node-codes derived previously - remember, an edge on the graph was defined by a pair of opposing faces on the cube. So an edge can be encoded with 8 digits by concatenating the codes for the two nodes it connects, sorted in lexicographic order (the edges are undirected, after all). And, a cube can be represented with 24 digits by concatenating the codes for the three edges it contributes to the graph, again sorted. These two sorting steps mean that we lose info to distinguish mirror image cubes, but as previously established, that doesn't matter to the solution!

Finally, the whole puzzle's 96-digit certificate is the four codes for the cubes, again sorted. And here again, the sorting doesn't matter, since the order of the cubes doesn't matter to the solution. For the both the Instant Insanity puzzle and the Schossow/Katzenjammer design, this certificate is (shown with some punctuation to aid understanding):

0221-0221  0230-1221  0230-1230  (cube #2)
0221-1221  0221-1230  0230-1230  (II#1/SK#3)
0221-1221  0230-1230  1221-1221  (cube #4)
0221-1230  0230-1221  1221-1230  (II#3/SK#1)

022102210230122102301230022112210221123002301230022112210230123012211221022112300230122112211230

My claim is that any (6,4,4) puzzle with this certificate is isomorphic to the original Schossow design.



This is a vintage advertising premium called Symington's Puzzle. It contains four cardboard cubes, each with a different arrangement of four Symington's product advertisements on their faces: Soup, Custard Powder, Ideal Cream, and Gravy. It is shown in Slocum and Botermans' "Puzzles Old & New" on page 38.
Isomorphic to SK.


The Great Tantalizer
Isomorphic to SK.


Tantalizer - Shackman #7415
Isomorphic to SK.


Here is another vintage cube matching puzzle called The FourAce Puzzle. The four wooden cubes are decorated with various arrangements of the four playing card suite symbols: hearts, diamonds, clubs, and spades. The box says "Provisionally Protected" but does not identify the manufacturer or date of manufacture. According to Slocum, this puzzle was sold in Britain at Gamage's in 1913.
Isomorphic to SK.


An advertising version of the Tantalizer puzzle, from Bass.

Taktikolor

Logi-Qubes

Those Blocks

Krazi Kubes

I am not sure what the left puzzle is actually called, but on the bottom of the tray it says "Masudaya Made in Hong Kong" so I call them the Masudaya Cubes. This may be the same as Ideal's Face Four puzzle, which I also found. Celia Seide notes that in Germany these are known as "Trikki 4."

Devil's Dice - Pressman

Devil's Dice - made in Hong Kong No.104A

Daffy Dots - Reiss 1971


Enigmus
Not isomorphic to SK.


Logicubes by Kaufmann / Galil
Isomorphic to SK.


"Can you solve those Damblocks?" were offered by the Schaper Manufacturing Company, Minneapolis, Minn. in 1968. I have examples in red, white, and black.

I got Mutando by Logika at Games People Play, and Mutando II from Time Machine Hobby.


Nice Cubes
Isomorphic to SK.


The vintage Cubo Color Puzzle.
Not isomorphic to SK.


Coloured Cubes
Peter Pan Series Regd.

Color Cubes

Crazy Blocks Color Puzzle
Created for Jak Pak Inc. of Milwaukee
Made in British Crown Colony of Hong Kong

Levenger Cubes
This vintage 1967 Instant Insanity clone by
"A to Z Ideas Inc." of California is called Psykonosis:

Here are the cube layouts:

  G      R      W      G  
R B G  B G W  W R R  G W B
  G      B      B      W  
  W      R      G      R  
                          
  1      2      3      4  
Here is the graph - Psykonosis is not isomorphic to SK!


Idiot's Delight - Field Mfg. Co. Inc. NY
Cube set signature 01143639 - Isomorphic to SK

Suits Me - Steve Kelsey
Arrange the four cubes in a row such that all four long sides show each suit once. (Eq. to SK)

There can be more than four cubes and four colors...

The Allies Flag Puzzle is another very old example of this family. This puzzle has five cubes, and each cube has some arrangement of five flags on its faces. According to Slocum, this puzzle was sold in Britain at Gamage's in 1915.

Dorobo - Hanayama

Hlavolam Iribako

This is Meffert's "Drives You Crazy." It includes six cubes instead of the usual four.
 
six cubes

The pieces aren't always cubic...

This is the Masudaya Hexagon Mind Exerciser. It has six hexagonal pieces. The objective is to line up the six hexagons so that each of the six rows of six faces shows all six colors. Unlike a set of cubes, where on each cube two faces are not used in a solution, here all six faces of every hexagon will be used. This means that there must be in total six faces of each of the six colors. In my copy of the puzzle, all six hexagons have distinct color arrangements - i.e. there are no duplicate pieces. I have found at least one solution - one can employ the graphical technique, but not in exactly the same way as for cubic puzzles - here, three mutually consistent sub-graphs are needed, and they are not independent.

A vintage Cylinder Ten by Masudaya.
Ten rotating disks each with ten color segments.
The English instructions say, "Arrange 10 different colors in a line to get all rows correct. The probability of getting the correct order is one in a billion. This puzzle will give the whole family hours of amusement. Note: it is packed in the factory with colors in the correct order (i.e. the solution)"
There is no date I can find on any of the material. However, I translated "Cylinder 10" into Japanese and did a search on Google Japan just to see what would turn up, and I found a "FrogPort" blog article from April 2008 talking about the release of this puzzle and how it is in a series with Masudaya's other puzzles Face 4, Hexagon Mind Exerciser, and Circus 7. Here is a link to the blog via google translation to English.
The instructions say "10 different colors in a line" but do NOT stipulate that the SAME ten colors must be in every line! This is unlike the traditional Instant Insanity type where one of each color must be present. This allows the rotating disks to have repeated or missing colors. A "line" is a row of stickers across all ten disks. There are ten disks each with ten positions/stickers, so they are accurate in saying one in a billion, since there are 10^9 possible settings (the setting of the first disk is immaterial).

Go Crazy
Embree Manufacturing Co. NJ 1969
Arrange the five disks so that alternate rows have five different colors, then three different colors. The five disks are separate and may be removed from the case and re-ordered.

The Buvos Golyok is a clever variant using balls enclosed in a tube.

Bognar Planets (Bolygok) - brown, white

Hungarian Tactics
Stan Isaacs sent me examples of his Doctor Octo and Dr. High Octane puzzles. These versions use octahedrons rather than cubes, and a clever mirrored base to allow one to see all faces. Thanks, Stan!


The cubes are usually labeled with colors, but sometimes numbers are used...

Twenty Teazer
Arrange the cubes so that each side totals 20.

Crazy Cubes

Tantalizing Ten - Shackman
Neurotic Numbers is a vintage 1968 puzzle from Lakeside

All sides must total 10 w/ no repeated numbers.

Here are the cube layouts:

  2      3      4      2  
1 4 3  1 4 1  3 4 3  2 4 1
  3      1      1      3  
  2      2      2      1  
                          
  1      2      3      4  
Here is the graph - Neurotic Numbers is isomorphic to SK!


Kathy's Kubes by R. Gee
Arrange so dots total 10 on each long side.

This type of puzzle can be arranged vertically, too...


Steiffel Tower

Totemania

Match 'Em High - Roads

Match 'Em High - Pipes

[59]

Some Solutions

I used the graphical technique to solve Nice Cubes:

Below is my solution to the Masudaya Cubes, using the graphical technique. The connectivity of this graph is similar to that of the Nice Cubes, though its edges seem differently labeled. Are the puzzles in fact isomorphic?

Explorations

So, why have so many puzzles over the years simply copied the topology of the Schossow-Katjenjammer cubes? Is it so difficult to find a different set of four cubes that satisfies the constraints and yet has only one solution? I set out to try to answer that question - here are notes on my explorations and findings.

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.
  1. yellow (24)
  2. green (16)
  3. teal (16)
  1. lavender (24)
  2. tan (10)
  3. 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.


I created a program to solve a 4-cube set, and used it within another program that explores 4-cube sets taken from the stable of 52 cubes above. I did not allow repeated cubes in a set - so, here again, I have simplified the problem space, but again I believe in a reasonable manner. I also eliminated repeated sets that differed only by the order in which the same cubes were selected, but this entails no loss of generality. This resulted in 270,725 sets being tried - a run that easily completed overnight on my PC. Note that 52_C_4 = 270725.

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!


I ran the 5160 single-solution sets through my program that computes a puzzle's certificate and checks if it is isomorphic to SK. I found that 24 of the 5160 are iso. to SK - these probably represent the sets that arise over the 24 color permutations for ABCD. If all four cubes in a set are permuted in a consistent manner, one may arrive at another set - and the two sets are in some sense the same puzzle.

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!

Here are the 24 signatures for the sets that share the SK certificate,
022102210230122102301230022112210221123002301230022112210230123012211221022112300230122112211230

01 04 20 40
01 14 36 39
02 03 22 45
02 12 33 44
03 07 17 40
03 14 26 47
03 30 35 38
04 08 16 45
04 12 28 32
04 29 37 46
06 20 33 47
06 22 32 39
07 10 28 30
07 14 44 48
07 18 39 46
08 10 26 37
08 12 36 51
08 18 33 38
12 20 25 35
13 17 33 37
13 22 30 51
14 22 25 29
15 16 30 39
15 20 37 48

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

Could these be the most difficult puzzles of the Schossow/Katzenjammer/Instant Insanity family?


The 5160 single-solution sets I found group into just 200 distinct certificates. I believe this reduction is also due to the permutations of the color assignments.

The certificate with the smallest number of sets having that certificate is shared by only six sets:
022022400220224022402240022022400220224022402240022022400220224022402240022022400220224022402240
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:
112011211121122112211221112012211121112112211221112012211121122112211221112012211121122112211221

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.

Wellingtons Cube Puzzles

An extensive set of additional puzzles in the Instant Insanity family were offered by three U.K. companies: Wellingtons Ltd. (they don't use an apostrophe), Onsworld Ltd. of Stamford UK, and Images & Editions. Many of them comprise four, six, or eight clear plastic cubes containing images on each side. In several cases the objectives are a departure from that of the Instant Insanity family.

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:

Bananas

Bananas 1982 Onsworld Ltd.
"Arrange cubes to show four complete bananas each with a label."
There are four cubes, and four types of side: a banana tip, stem, middle segment with no label, and middle segment with label. Unlike a standard II-type puzzle, the images have distinct orientations which must be respected to form the complete bananas. The tips, stems, and middle segments must all be aligned properly. A little analysis reveals that the usual four-in-a-row arrangement of the cubes cannot satisfy the goal - but then, the objective does not stipulate that arrangement, does it?
Bananas II

Bananas II
(I don't have this one.)
Blue Movie

Blue Movie - 1986 Wellingtons
I don't have this.
Body Job

Body Job - Onsworld
I don't have this.
Boob Cube

Boob Cube - Wellingtons
I don't have this.
Booby Trap

Booby Trap - 1986 Wellingtons
I don't have this.
Bunkered

Bunkered 1987 Wellingtons
"Tee-off by placing the cubes together then rearrange them to show, simultaneously, TWO identical golf courses."
The Cat Puzzle

The Cat Puzzle - Wellingtons
I don't have this.
Image from Jim Storer's collection.
Computer Challenge

Computer Challenge - 1984 Wellingtons
I don't have this.
Computer Magic Square Challenge

Computer Magic Square Challenge
1995 Images & Editions
Computer Word Challenge

Computer Word Challenge - 1985 Wellingtons
I don't have this.
Crossword Cubes

Crossword Cubes - 1989 Wellingtons
I don't have this.
Here is a possible alternative package:
Cubix 3D

I don't have this, but see Rebus in the Pattern Blocks section below.
Cuss

Cuss 1980 Onsworld Ltd.
"Simply arrange the blocks in a row so that each side carries a dictionary four letter word, all reading from left to right. No foreign words or proper names allowed. A clue in case you get stuck and a solution are included."
"This is one of a unique range of cube puzzles devised for Onsworld by Stephen Leslie. The series includes Cuss, Diabolical, Footsie, Frantic, Son of Cuss, and Walk Up."
Son of Cuss

Son of Cuss 1982 Wellingtons
Devised by Stephen Leslie
"Arrange the cubes to show at least 24 different four-letter (dictionary) words."
Diabolical

Diabolical (Onsworld)
"Arrange the dice in a row so that the four long sides all carry the same total number of spots."
Multiple solutions exist - the set of addends doesn't have to be same for each side.
Double Cross

Double Cross
Wellingtons 1991
Fascinating Felines

Fascinating Felines - 1993 Wellingtons
Footsie

Footsie 1982 Wellingtons
"Arrange the four cubes in a square so that wherever the surfaces of two cubes join (top, bottom, and sides) pairs of feet are formed."
Devised by Stephen Leslie.
Frantic

Frantic 1980 Onsworld Ltd.
"Place the four cubes in a row and simply rearrange them so that one of each colour shows on each long side."
Devised by Stephen Leslie.
Isomorphic to SK!
Frantic (alt.)

An alternative version of Frantic, with solid colors. Colin says it is the same graph.


Also issued with black backgrounds.

Frantic II

Frantic II
"Place the cubes together in a square so that wherever they meet, top, bottom, and sides, the coloured squares match."
Frantic II was invented by Dr. Kenneth Miller.
Golf

Golf - Wellingtons
"Places the cubes together to show the word GOLF four times. The letters must all point the same way. Try it - then baffle them at the 19th."
Golf Crazy

Golf Crazy - Wellingtons
Grandstand

I don't have this.
The Great British Puzzle

The Great British Puzzle - 1997 Images & Editions
The Kitten Puzzle

The Kitten Puzzle - 1999 Images & Editions
by Dr. Kenneth Miller
Knickers

Knickers - Wellingtons
I don't have this.
"Arrange the knickers in a line so that there is one of each colour on each long side - they can be in any order and face any direction."
Marilyn - The Eternal Puzzle

Marilyn - The Eternal Puzzle - 1989 Wellingtons
Match of the Play

Match of the Play 1989 Wellingtons
"Solve the puzzle by placing all six cubes together so that both sides show a player being tackled."
Monkey Business Spring Fever

Monkey Business Spring Fever - Wellingtons
I don't have this.
Nuts

Nuts 1982 Wellingtons
"Place the four cubes in a row and simply rearrange them so that one of each kind of nut shows on each long side."
Isomorphic to SK!
180 Top Dart

180 Top Dart 1990 Wellingtons
"Place the cubes in a straight line so that all four long sides add up to the magic number... one hunder and eighty!"
There are six cubes, and the following 12 symbols appear: inner bullseye (50 pts), triple 2 (6 pts), double 3 (6 pts), triple 4 (12 pts), double 6 (12 pts), 9, double 9 (18 pts), triple 9 (27 pts), 12, double 17 (34 pts), 18, triple 20 (60 pts).
The Pig Puzzle

The Pig Puzzle
Images & Editions
Poker Puzzle


Poker Puzzle 1987 Wellingtons
"Place the dice in a line so that one of each design appears on each of the four long sides - the order and directions of the designs are not important."
There are 5 cubes. The designs are: 10, Jack, Queen, King, Ace.
Rugby

Rugby - 1995 Images & Editions - Dr. Kenneth Miller
Two puzzles based on the two versions of Rugby - for the Rugby League fan, place the cubes in a straight line so that each long side shows exactly 13 pieces of muddy lace. For the Rugby Union fan, place the cubes in a straight line so that each long side shows exactly 15 pieces of muddy lace.
I don't have this.
Seams Impossible

Seams Impossible - 1986 Wellingtons
I don't have this.
"Place the cubes in a square so that wherever two cubes join (top, bottom, and sides) complete legs are formed."
Snookered

Snookered - Wellingtons
"Place the cubes together in two rows of four, to make a snooker "table." Then rearrange them to show three sets of snooker balls: one on the top, one on the bottom, and one on the four sides of the "table." A set of snooker balls comprises: 15 reds and one each of white, black, pink, blue, brown, green, and yellow."
Snookered Again

Snookered Again - Wellingtons
"Arrange the cubes in a block (two rows of three) so that one, and only one, of each of the eight colours shows on the top, sides, and base of the block."
Soccer Balls (?) (Football)

No packaging - we're assuming these are Wellingtons
Please email if you have information about this puzzle. Thanks!
Soccer

Soccer Puzzle - "Make it your Goal" -
Wellingtons 1987
Spellbound

Spellbound
Spellbound (alt.)

Spellbound - alternative version - Onsworld Ltd. 1980
Suit Yourself

Suit Yourself 1990 Wellingtons
"Place all six cubes together in a block of two rows of three to show (on top, bottom, and sides) a full pack of 52 cards with no duplicates."
Tantalizer
Crossword

I don't have this. This and the other Tantalizer puzzles seem to be more dexterity than Insanity-type.
Tantalizer
Frustration

I don't have this. (Photo from Slocum collection.)
Tantalizer
Jet Set

I don't have this. (Photo from Slocum collection.)
Top Dart

Top Dart
Total Distraction

Total Distraction 1988 Wellingtons
"Put the cubes together to form one large cube with all rows, columns, and diagonals having the same total: 99."
Walk Up

Walk Up 1980 Onsworld Ltd.
"Place the four cubes on top of each other in a column. Arrange cubes so that ladybird (sic) tracks reach from top to bottom on all four sides. Ladybirds need not point all the same way."
Devised by Stephen Leslie.
Watch It

Watch It - 1998 Images & Editions
I don't have this.
World Puzzle

World Puzzle - Wellingtons
I don't have this.

[22/53]

Waddingtons Mindbender Puzzles

Here is a series of puzzles issued by House of Games Corporation Limited of Bramalea, Ontario, Canada. They're all made of sturdy cardboard. Some of them are shown elsewhere on this site.

Per the pamphlet that came with the Rectangle Tangle puzzle, John Waddington Ltd. of Castle Gate, Oulton, Leeds England also made these under license from House of Games.

Waddington was bought by Hasbro in 1994 for 50M pounds. Read an article about Hasbro's spree.

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.

The most frequently mentioned, most dating from 1969, are:

  • The Perfect Square (1969)
  • The Tricky Triangle (1969)
  • The Perfect Circle (1969)
  • The Wobbly Web (1969)
  • The Perplexing Pyramid (?)
  • The Path (1969)
Less often mentioned, dating from 1970 and 1971, are:

  • The Reluctant Rectangle (1970)
  • Little Circles (1970)
  • The Coloured Square (1971)
  • Rectangle Tangle (1971)

Also mentioned once is "The Coloured Pyramid" which I believe is the same as The Perplexing Pyramid.

The brochures also mention Beat the Elf, Kolor Kraze, and Cube Fusion.

The Path
1969 #505

Form a 3x4 rectangle using 12 colored square tiles which are each printed at from 1 to 3 of their corners with small circles. There are 4 tiles each of 3 different colors, blue, orange, and pink. All pink must be in the first row, all orange in the second, and all blue in the 3rd. One circle on a pink tile is marked START and must be in the upper left, and one circle on a blue tile is marked FINISH and must be in the lower right. Create a path of circles from START to FINISH such that circles are not adjacent unless they form a path connection.

 

The Perfect Square
1969 #505
Assemble a square using 12 pieces - like colors must not touch.

The Tricky Triangle
1969 #505
Using 10 barbell-shaped pieces composed of two linked circles each, and one single circle, build an edge-6 equilateral triangle (of 21 circles) such that "no two circles of the same color are in the same line of the triangle." Each circle is 1 of 6 colors.
The Tricky Triangle puzzle by Waddingtons is a 2-dimensional analogue of Oops Again.

The Perfect Circle
1969 #505
Assemble a circle from 16 pieces in 4 basic shapes and 3 different colors, such that like-colored pieces don't touch.

The Wobbly Web
Copyright 1969 No. 505
Create a rectangle from the 15 square tiles such that web strands join (edgematching).
JH Vol.2 p210

The Perplexing Pyramid
14 square tiles each of which has a colored circle in each corner. Create a "pyramid" of cards - a 3x3 base, then a 2x2 layer, then one card on top - layers must align over spots, and spot colors must match vertically.

The Reluctant Rectangle
1970 #345
Form a rectangle from the 12 3x2 L-shaped pieces such that no two pieces of the same color touch in any way. There are 3 pieces each of four colors. All the pieces are "normally-handed" L's except for two of one color are reversed.

Little Circles
1970 #346
Pack 18 pieces formed from circles into the box. (The instructions are printed on the box, and there is no mention of a color-dismatching constraint.) No Pamphlet.

Coloured Square
1971 #340-E
Form a square from 12 L-shaped colored tiles such that like colors don't touch.

Rectangle Tangle
1971 #340-F
Form a rectangle with a bi-color perimeter from the 10 tri-color and 2 bi-color L-Trominoes.

Beat the Elf - by House of Games Corp. Ltd. Don Mills, Ontario 1970
Build a 3x3x3 cube with the 13 blocks so that no face shows three squares in a row of either color, horizontally, vertically, or diagonally. There are 11 1x1x2 blocks of 1 light and 1 dark. There is one 1x1x2 block of two darks. There is one 1x1x3 block of two lights (adjacent) and 1 dark.

Kolor Kraze - by House of Games Corp. Ltd. Don Mills, Ontario 1970
Discussed in Slocum and Botermans' Puzzles Old and New on page 43. The Kolor Kraze puzzle consists of 12 dicubes and a tricube, each made of differently colored cubes. There are 3 each of 9 different colors. As in the Gram's Cube, the goal is to create a 3x3x3 cube from the pieces, such that each of the 9 colors appears only once in each of the six faces of the cube. Sivy Fahri wrote an article about the Kolor Kraze and the Nine Color Puzzle called "Nonahuebes" he developed based on it. A PDF of Fahri's article is available at the Gathering 4 Gardner Wiki.

Cube Fusion
Alright, it's a game not a puzzle, but I remembered these pieces from my childhood and now I know what they belonged to!

Other Color-Constraint Puzzles

There are several other puzzles, cousins to Instant Insanity rather than siblings - which involve some kind of color constraints.


Gram's Cube was made by Gram Toys of Birkerod, Denmark. The puzzle consists of 27 Lego-like cubies that mate side-to-side as well as up-and-down. There are 3 cubies each of nine different colors. The objective is to construct a 3x3x3 cube such that each side shows all 9 colors. At first I thought a trick was necessary, but I found a solution using all 27 cubies. I picked this up in a trade with Norman Sandfield, at the January 2005 New York Puzzle Party.

Level Q, by Eng's IQ Co. Ltd. 1987 Hong Kong. I purchased this quite some time ago. Level Q consists of a hexagonal board and twelve bar-bell shaped pieces. There are three challenges - first, build seven stacks of six disks each. Next, again build seven stacks of equal height, but such that one bar lies on each side of the hexagon and on each of the six spokes. Finally, satisfy the constraints already mentioned, and also ensure that each stack contains only one of each color disk.

The Trapagon
and Magzphere
Six pieces interlock - arrange them so that there are five different colors on each "face."

The object of Oops Again is to build a pyramid with the 2-sphere pieces so that no two spheres of the same color touch at all. The Golf Smarts Pyramid (a gift from Brett) is similar.

Spot Cube - Hikimi
Designed by Ichiro Sengoku, this puzzle was entered in the 2004 IPP Design Competition. I bought it at Torito.
  There are 3 challenges:

1. Position or pile the cubes to show only the 6 green spots and no others.
2. Show only the 8 yellow spots.
3. Show only the 4 red spots.
Spots face-down on the table are considered hidden.

 

Vier Farben Block - Logika
Build a cube with the 12 u-shaped pieces such that like colored pieces do not touch. Each piece is one of four colors.

Instant Indecision
marked "Patent Pending" - the Green Gate Co. Sherman Oaks CA, 1972
Four cubic frames and a stepped pedestal. Each cubic frame contains twelve bars, 3 each of four overall different colors, arranged in various patterns. The four cubes are of graduated sizes and nest on the pedestal. The objective is to arrange the cubes in nested form on the pedestal so that each of the twelve sets of four aligned bars, one from each cube, contains one of each of the four colors. My version has a nice key-like tab that locks into the pedestal and holds the cubes in place for storage/transport.

Cube Edu - eLogIQ
Build a 2x2x2 cube and match colors on all faces.

Octamania
Arrange the 8 cubes into a 2x2x2 cube such that each face shows four copies of a digit.

The vintage 8 Blocks to Madness puzzle - invented by Eric Cross of Ireland, and issued by Austin Enterprises of Ohio.
Create a 2x2x2 cube having a single color on each side. Then create a 2x2x2 with each of four colors on every side.

The Maze Cube -
stack the four bricks together to make a 2x2x2
having four colors on each face.

Poker Cubes
I don't have any documentation with this puzzle, so I don't know the provenance or goal. However, I suspect the goal is to create a 2x2x2 having one of each of the four suit symbols on every side, while also ensuring each 2x2 face is a single color with each face a different color.

The Ten Spot Domino Puzzle
Issued by Valentine & Sons, Ltd. Westfield Works Dundee
The instructions inside the box top read, "Valentine's Series of Popular Puzzles
The New 10 Spot Domino Puzzle
Instructions - Arrange the four pieces so that they form a solid
square block, every side of which shows 10 spots.
Solution sent on receipt of 1d. stamp."

Discussed by Slocum & Botermans in their 1994 book The Book of Ingenious & Diabolical Puzzles, on page 34. The layout of the pieces and one solution are provided on page 143.
Analyzed by Len Gordon and found to have seven solutions. Len also found that the pieces could be arranged to show 11 on all faces in two ways.

I finally found a Cover Up - issued by Ideal in 1982, it originally appeared as Hepta in 1974 and was designed by the famous Alex Randolph. See the entry for Hepta at Boardgamegeek. I had wanted to try this puzzle since reading about it at Celia Seide's website. Celia also pointed out Spielarchiv.de which has a reference to a version called Magic 7 under Spiele->nach Autoren->R->Randolph,Alex -> Magic 7.
There are seven plastic 1x3 straight pieces and seven angle pieces. The plastic 7x7 board is colored using 7 colors each of which appears 7 times. For each of the 7 colors, use all the pieces to cover all the spots not of the target color. It's pretty large - I put a quarter coin on the cover.


Cubic Mania Puzzle Blocks was issued by Dale Seymour Publications. It comprises eight cubes, each colored with four colors. Each cube uses each color at least once. Using the chart of cube graphs I developed, the included cubes can be mapped to the tan set { 24 x 3, 27, 34, 42, 43 x 2 }.

There are several challenges, including building a 2x2x2 where every side shows all four colors, and another where every side is a solid color.


Brain Wave
Arrange the bi-colored pieces in the slots
so that no color is repeated along any line of the
triangle on both sides of the head.

Lucky 13, Edition No. 7, by Fireside Games Inc. of Northbrook Illinois, 1973.
Comes with 38 translucent plastic pieces, each being one of 5 colors and 13 different shapes composed of unit squares joined by sides (square polyominoes) - each shape has a letter ID.
The table below gives the shapes, grouped by color. The color cell gives x @ y, the count x of each piece type and the number y of unit squares the piece occupies.

Red
6 @ 2
Orange
6 @ 3
Green
6 @ 4
Yellow
2 @ 5
Blue
2 @ 6
A B F J L M Q S T V Z CC FF
[] []
[] [] []
  []  
[] [] []
    []  
[] [] [] []
  []  
  []  
[] [] []
    [] []
[] [] []  
[]   []
[] [] []
[] []
  []
  []
  []
  []
[] [] []
    []
    []
    []
  []  
[] [] []
  []  
  []  
[]  
[] []
  []
[] []
[]   []
[] [] []
  []  
  []
[] []
  []
  []
  []

24 different puzzles - in each, arrange 13 pieces to form a 7x7 square such that like colored pieces do not touch, even at the corners. Any piece may be flipped.
Always use 3 x A, 3 x B, and 3 x F.
For the remaining four pieces, pick a puzzle number from 1 through 24 and find it in the chart below.
Each row and column is labeled with a piece ID.
Use two copies of the piece identified by the row in which the puzzle number appears,
and two copies of the piece identified by the column in which the puzzle number appears.

  S T V Z CC FF
J 1 2 3 4 5 6
L 7 8 9 10 11 12
M 13 14 15 16 17 18
Q 19 20 21 22 23 24

MacMahon Colored Cubes and the Mayblox puzzle

Percy Alexander MacMahon was a mathematician who lived from 1854 to 1929. He is noted for, among other accomplishments, his results in the field of combinatorics. In 1915 and 1916 MacMahon produced a two-volume treatise on Combinatory Analysis which remains a respected work today. MacMahon also produced New Mathematical Pastimes, published in 1921. This work discusses his Colored Cubes, first introduced in a lecture he gave in 1893 [4].

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.


References:
  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

Pattern Blocks

There have been several issues of sets of "pattern blocks" under various names.


Pattern Pending designed by Fred Horowitz and issued by Parker Brothers (General Mills) in 1971. A set of 12 black and white cubes. Fred had the basic idea for these in the 50's, and intended these to be more of an open-ended creative playset than a puzzle.

K-Dron - Janusz Kapustra
(I don't have this.)

Trac 4 game issued in 1976 by Lakeside. Red-and-white blocks and cards specifying various patterns to construct. (Sadly, this got lost in the mail and I never received it.)

Quiet Cubes
(I don't have these.)
Also designed by Fred Horowitz, a set of 18 cubes, issued in Holland.

Rebus
A set of nine patterned blocks, issued by the German company "Pussy." Create various silhouettes.

Magritte Cube

Pattern Puzzles Using Mirrors


Tedco Mirage Numbers Up

At Games People Play, I found a Magic Mirror puzzle from Schmidt Spiele.

Escher Blocks from RecentToys, purchased from Bits & Pieces.

Puzzling Reflections
Ivan Moscovich
1990 Tarquin Publications

The Mirror Puzzle Book
Marion Walter
Tarquin 1986
ISBN 0906212391

You and Mona Lisa Mirrorkal
Invented by Ivan Moscovich
Produced 2011 Recent Toys USA

You and Einstein Mirrorkal
Invented by Ivan Moscovich
Produced 2010 Recent Toys USA

Other Pattern Puzzles


This is the "Fool's Spool" (made in Hong Kong). "Mix up the four wheels - rearrange them so that all lines total twelve!" One of my kids found the following solution in minutes: 5115, 1254, 5223, 4341, 2334, 3513, 4422, 3333.

Here are more puzzles with a math theme. One is "Digi-Disc" - arrange the discs so that all equations are true. The discs adhere via magnets and I also show this puzzle in the Magnetic category. Another, from Asia, is a similar stack of discs with the same goal, but it is not magnetic. Yet another is an elongated stack, non-magnetic, called "Magic Numbers."

Sudoku 3D Ball - 22 Advance
www.3d-jp.com
There are 5 circular indents on each pentagonal face of the spherical dodecahedron, and 10 indents in a "ring" bordering each face. Each indent is a member of 2 "rings" - those around the two faces adjacent to the vertex where the indent is located. Some indents already contain a digit from 0 through 9. Fill in empty indents with digits from 0 through 9 such that every ring contains all 10 digits.
(The sphere is not meant to be disassembled.)

Alpha Snake 9 - designed by Ken Irvine
from Creative Crafthouse
Rotate the blocks until each of the four long sides shows the letters A through I (with no repeats).

Math Snake 9 - designed by Ken Irvine
from Creative Crafthouse
Rotate the blocks until each of the four long sides shows a valid equation.

A Washington Monument "sums" puzzle, from Lambert Bright. Thanks!

Kenken 4 - in maple, from the kind folks Adele and Peter Plantec at Puzzle Me Please.
Arrange 1x1x2 blocks in the frame to form a 4x4x4 cube such that a valid KenKen 4 appears on each face. The blocks only have figures on the outside.

Kenken 6 - in maple, from PuzzleMePlease.com
Arrange 27 cubelets in the frame to form a 3x3x3 (6x6x6) cube such that a valid KenKen 6 appears on each face. The cubelets only have figures on the outside - the central cubelet has none. So, all figured faces will be used in the solution.
 

The Enigma puzzle is a representative of this type, but with letters rather than numbers on the wheels. Form words.

This is the Daily Mail Crossword Disc puzzle by Chad Valley.

Daily Mirror Double Words

There are several "Slivers" puzzles - this is the Anakin Skywalker version.

Zen Blocks

Cubits

That-A-Way by Binary Arts. Introduced at IPP 20 by its creator Greg Dye. There are 10 cards/tiles each with two arrows in various orientations, and a booklet of problems and solutions. Play That-A-Way on-line.

Flower Finder - William Waite
Arrange the six pieces in the tray so that every flower has six petals. Also solve so that every flower has three petals.

Rubik's Dice
flop interior plates around until all pips show red.

Binary Arts, now Thinkfun, has supplied many puzzles, including their various "something by something" puzzle/games:
Brick by Brick, Square by Square, Block by Block, and Shape by Shape.

Dice Stacker by Elverson
Eight wooden dice, each with a unique arrangement of spots - form a cube having four of a kind (one through six) on each side. Also, form a cube such that diagonals on each side add to seven.


Chainbox
1981 Yonezawa - made in Japan
Purchased at IPP28
A case containing four identical "chains" each made of eight black plastic links. Some of the links have orange shapes on them, front or rear. Includes a booklet of patterns to be made in the tray, using the four chains folded and arranged in different ways.


Pakostar
Purchased during IPP28 in Prague

Ubongo
Purchased in Berlin on the way to IPP28 in Prague.
Really a game for 2-4 players, but can be played solitaire. Given a set of pieces and a deck of cards showing shapes made from those pieces, take a card and try to arrange the pieces to match the silhouette.

Identica-L episode II
Concept by Tadao Kitazawa, 2008. Purchased from Torito.
Four pieces, each of which has a pattern of light and dark squares on each side. Arrange the four pieces so that the light squares form a connected shape, and the dark squares form the same shape simultaneously. The pieces can be flipped.
See analysis at Ishino's site.

Hinomaru by Pavel Curtis
Build the Japanese flag in the tray, using the 12 two-sided domino pieces.

Anansi's Maze by Pavel Curtis. Purchased at IPP 29 in SF. Also available via Pavel's website.
A multi-stage challenge.

The Blockword Crosspuzzle from "Products of the Behavioral Sciences" ca. 1970

Gamesters of Triskelion,
a multi-stage challenge by Pavel Curtis.

Tiger and Panther - designed by Takahisa Nakanishi - IPP30
There are four T-tetromino and four P-pentomino pieces, each checkered.
Pack them in the 6x6 tray to make stripes, and then to make a checkerboard.

Figure Puzzle - designed by Hide Ue
presented at IPP30 by Naoyuki Iwase (Osho)
Seven pieces, colored light and dark variously on both sides.
Arrange the pieces in the tray to form each digit 0-9 in turn.

Tridio Twist! by FatBrain Toy Co.
Three mutable pieces and 48 pattern challenges. Designed by Marijn van Herel and Eliane Scharten.

Designer Colin James very kindly sent me a copy of his Qboid puzzle.
It offers over 1000 challenges at four different levels. 12 cubes, with patterned faces. When properly arranged, the patterns form digits and letters. Entered in the 2009 IPP Design Competition. Thanks, Colin!

Royal Flush
Arrange the seven L trominoes and 2 dominoes in a 5x5 grid such that every row, column, and both main diagonals contain a Royal Flush (in spades). Thanks, Dave!

Dewey's Dilemma
From Dave Janelle at Creative CraftHouse.
Arrange 12 consecutively numbered "books" - each being one of three heights, and by one of three authors - on the included "bookshelf" such that no adjacent books are the same height, or by the same author, or consecutively numbered.

PrimEvil, designed and made by Louis Toorenburg, exchanged at IPP32 by Paul Dudding

Liberty Cubes, designed and exchanged at IPP32 by Kate Jones, made by Kadon Enterprises Inc.

Double D Insanity, designed, made, and exchanged at IPP32 by Louis Toorenburg

The Bricklayer's Challenge, purchased at IPP32 from Pavel Curtis

Nats Win!, designed in 1938, made by Creative Crafthouse, exchanged at IPP32 by Michael Tanoff

Bendit - from Smart Games.
Designed by Raf Peeters.
I purchased this at a Barnes & Noble store near me.
Six different pieces, each having a sequence of black and white segments and hinged in two places. Based on a booklet of 60 challenges, fold and place the pieces in the 6x6 tray to compose a target pattern.


Ji-Ga-Zo - website - issued by Tenyo in Japan and Hasbro in the US. Designed by puzzle-friend and magician Mark Setteducati.
Kind of a color-by-number set. 300 pieces, in various shades of a color. Use supplied software to input an image and receive instructions on how best to arrange the pieces to produce the image. Similar to the art of Ken Knowlton.

J Puzzle, designed and exchanged at IPP32 by Tsugumitsu Noji, made by ASUNARO
Four pieces, three challenges: make a symmetric shape from 3 pieces; make four holes with an arrow pointing at each hole; make outlines of two pentominoes.

Noah's Ark - designed by Raf Peeters, issued by Smart Games
My daughter's favorite. Place the pairs of animal pieces in the ark, ensuring that the animals are upright and mates touch. Each challenge specifies the required position of one or more pieces.

Pinwheel Puzzle - produced by Creative Crafthouse
Arrange the tiles so that all nine stacks including numbers on the rim add to 20.
Alternately, arrange so that all nine stacks excluding numbers on the rim add to 15.

Logic Dots from Brainwright is a pattern and logic puzzle - nine one-inch cubes with colored sides (one cube has six gold sides) must be placed into a 3x3 tray according to one of 36 challenge cards, each of which gives a series of clues. Eight of the cubes have three colors and opposite faces are colored the same.
The tray is made with a sliding clear plate on each side. When playing, remove one plate but leave the other in - for later challenges not only will you be rotating the tray, >but you have to flip it over and observe color positions on both sides of the tray.
You must deduce the correct cube positions based on the clues and ultimately locate the golden cube. At the outset of each challenge you are told the relevant set of 9 colored cube faces to be used "face-up" in the grid - since a cube's opposite faces are the same color, those will be the colors visible on both sides of the tray.
The logical challenges start simple but get much more difficult.
You may be reminded of Thinkfun's Chocolate Fix puzzle, but Logic Dots really ups the ante.
I think this would also be a good puzzle for precocious youngsters - at least the initial challenges. The box says for ages 8+.
Logic Dots was created by the Swiss team Atelier Rohner and Wolf.