More about edgematching

The idea of fitting matching or complementary tiles together is as old as dominoes, movable picture blocks, and jigsaw puzzles.

The concept of colormatching tiles based on all the permutations of colors on their edges dates from 1921, when the British mathematician Major Percy MacMahon invented and introduced Three-Color Squares and Four-Color Triangles in the first edition of his book, New Mathematical Pastimes.

MacMahon divided squares and triangles into triangles to give each edge of a piece its own color (see Kadon's Multimatch I and Multimatch III), in all possible combinations. Each set contains 24 different tiles, and MacMahon discovered that they could form a single figure with all adjacent edges matching and just one color all around the outside border. Other adjacency conditions can also be explored.

Scott Nelson designed a set of four cubes, whose faces were covered by the 24 Multimatch I tile patterns so that the colors matched on all 12 edges of each cube. The four cubes could then be matched to each other in various configurations. This puzzle was published by Binary Arts under the name of — what else? — 4 Cube Puzzle, now sadly out of print.

The Addicting Games website has a delightful online matching puzzle with edge-colored squares that lets you rotate and exchange tiles and even swap segments. They call it Kaleidoscope. It's simply beautiful and frequently infuriating.

The most extensive research into these sets, over three decades, was done by Wade Philpott of Lima, OH, who identified all the possible symmetrical shapes that MacMahon squares and triangles could solve with both matching edge colors and uniform border color, and who calculated all the numbers of solutions for the MacMahon Squares' 4x6 rectangle. Here's a hidden solution. These findings are included in the rule books for Kadon's Multimatch I and Multimatch III.

Another extensive researcher is Toby Gottfried, whose pioneering work in computer-derived solutions for the corner-colored squares (our Multimatch II) won him an honors degree in computer science in 1972. A version of this puzzle was made back then by Skor-Mor under the name, Nitty Gritty. Toby's program back then took 500 hours to accomplish what his 1999 computer can whip out in 2 minutes. You can see his results and play some puzzles and games on his joyous website. Toby's latest research explores wrap-around solutions for Multimatch II's 3x8 and 4x6 rectangles. For the latter he has found 180 cylindrical solutions. One is shown above.

Bernhard Wiezorke of Germany designed a charming matching puzzle he calls Matchpoint, which combines both corner matching and side matching, with 20 well-made blue cardboard squares with half-circles and quarter-circles in white and yellow. Getting them into a 4x5 rectangle is no mean feat. It does not appear to be in production beyond the sets he made for an International Puzzle Party in 1994.

Mag-N-If's plastic puzzle, On the Level, is a multi-level subset of MacMahon's squares, consisting of the 9 pieces that have three different colors (heights). The pieces look like little skyscrapers in a 3x3 array.

Pressman's Triominoes use triangles with edgematching of numbers, though they are not entirely combinatorial.

DaMert Company's Triazzle is a popular line of hexagonal and triangular cardboard puzzles by nature artist Dan Gilbert, with figures where heads and tails need to be matched up correctly. Triazzle puzzles and most jigsaw puzzles are meant to fit into only a single design. Combinatorial edgematching sets, on the other hand, have unlimited possibilities for play and pattern-making.

Eric Solomon has defined a set of 19 two-color hexagons with vertex colors that create curvy color flows and patches. You can try them out on his website's Hexagrams page and even see a gallery of fun and fanciful designs. The pattern at left is nicknamed "PingPong." Designs with perfect opposite-color symmetry (Eric calls them antisymmetric) are surprisingly easy to do. Here's a nice little edgematching game, Trinhex, you can play for free online with tiles like Multimatch IV. The programming is very fine, with easy and ascending levels of complexity.

A special subset of the edgematching genre is path, pipe or track connections. Commercial games like Trax by New Zealander David Smith; Connections; the Amazing Labyrinth; Binary Arts' golf course puzzle, Golf Tease; computer games like Pipe Dream and Entanglement; and Maureen Hiron's Continuo are of this type.

Kadon's growing collection of purely combinatorial edgematch tilings includes several sets with path formations, notably Kaliko, Arc Angles, Dezign-8, and Octiles.

Kurt Van Ness has a nice webpage showing Kaliko and other path-marked hexagon sets, including his own Serpentiles and the commercially popular Tantrix.

The number of tile patterns for any defined theme can be calculated systematically. Peter F. Esser in Germany has analyzed, collected and solved an awesome array of polygons with both color edges and contoured edges. His solver programs handle sets with up to thousands of unique tiles. Wander through his well-organized website, Peter's Puzzle and Polyform Pages, and be astonished. He covers many of the polyform and edge-match sets Kadon makes and extends them into realms only computers can follow. Huge solutions shown are dizzying.

An assembled set lends itself to a huge variety of surprising and attractive designs, and each set will have its own unique visual character. They are "movable art" as much as puzzles.

Noteworthy in this artistic category are Kadon's Triangoes and Triangoes Jr., which combine the characteristics of both edgematching and shapefitting, consisting of more than one shape (order 1 and 2 polytans) with color differentiations.

A two-volume Compendium of Card Matching Puzzles was published in 1995 by Jacques Haubrich of the Netherlands, with a 1996 supplement, About, Beyond and Behind Card Matching Puzzles. The Compendium covers over 1000 puzzles, divided into 7 main types, with 136 subcategories. For all this wealth of information, it is by no means exhaustive; periodic updates are planned as new information becomes available. These books can be purchased by contacting us. We'll forward your request to Jacques.

Zdravko Zivkovic expanded the 3D-matching concept by putting six colors, four at a time, on 90 all-different corner-colored squares that match along all edges on the faces of 15 cubes, which Kadon introduced in 2001 as Cube Brick in a limited first edition (sold out, currently out of print). Zdravko's Color Up cubes, also offered by Kadon, have three colors on a face, as two single squares and a double stripe. Still to come is Zdravko's Magic Carpet, a set of 15 cubes whose 90 faces are covered by edge-colored squares, using six colors, four at a time.

In 2007, Zdravko Zivkovic created an octagonal edgematching set, Doris, published by Kadon in 2008, with its world premiere at the Gathering for Gardner VIII in Atlanta. With all the combinations of 3 colors on their edges, the Doris tiles can be matched four different ways—an innovation not seen before. Doris won best puzzle of the year honors from Games Magazine for 2009.

Zdravko followed this triumph two years later with a hexagonal edgematching set, MemorIQ, which snagged a place on the Games 100 list for 2012.

Also in 2001, Kadon introduced the Fine Touch Collection of puzzles with tiles whose edges or corners are indented to create variously shaped matching symmetrical spaces. These puzzles are particularly suited for solving by touch only, a boon for visually impaired or blind players.

In 2002, Kate created a limited-edition 9-piece edgematching set just for the International Puzzle Party, using four differently shaped cut-outs instead of colors, Four on a Match, that has only two matched solutions. One of them is shown here. This set was inspired by and is the equivalent of MiniMatch. In 2003 it was slightly redesigned and became a full-fledged product in its own right, also bringing the MiniMatch concept to blind players while presenting an attractive visual puzzle to everyone.

Filigreed cut-outs also showed up in a set Kate designed for the 50th anniversary of SIAM, and in 2009 they became her own birthday souvenir puzzle, Celebrating LXX. With four types of edge markings—wide triangles, tall triangles (the golden triangles), circle and blank—the set encompasses 70 distinct tile patterns. The patterns can be cut-out or simply imprinted.

For 2011, Kadon finally published Kate's 25-year-old design, Fractured Fives, that has just 5 wood blocks with a different puzzle concept on each side:  matching twisted ropes on one side, and fragmented pentominoes on the other.

In 2012, Chris Hoffmann of Purdue University custom-ordered a 70-piece four-color version of the MacMahon Squares/Multimatch I, and we happily constructed the Grand Multimatch I set for him.

In 2013, Jared McComb custom-ordered a set of 28 V trominoes with embedded arrows, each tile with a head and a tail in every combination, to match up into complete arrows (right). Unsolved—can the arrows form one continuous loop that visits each tile exactly once?

There's no end to the emergence of new ideas for sets of all-different pieces that can happily co-exist in matching harmonies. Stay tuned!

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