The Games Journal | A Magazine About Boardgames

Polyforms

Kate Jones

February, 2001

It should not be controversial to say that "puzzles" may be defined as something to solve, to figure out, to deduce, to discover an answer to for the sheer fun of it. Puzzles are recreational thinking, and they have been a part of human culture for a very long time.

For hundreds, no, thousands of years, there have been riddles—word, logic and conceptual puzzles—such as the one ascribed to the Sphynx, about the creature that walks on four legs in the morning, two legs at midday, and three legs in the evening. Many riddles were embedded in rhymes, playfully disguising answers in metaphors and analogies. Such puzzles did not require any equipment other than wits and ingenuity.

The earliest puzzles that were some form of apparatus were probably locking mechanisms, like the complex block systems purported to have been used to seal the pyramids, and the Chinese rings used to lock treasure chests. The latter were so time-consuming to solve that any thief would have been discovered well before he could succeed. Rope tricks, like the legend of the Gordian knot, and embedded objects, like King Arthur's sword in the rock, were mysteries that verge on magic in the minds of the tellers of those tales. Antiquity abounds with elaborate puzzles, like Stonehenge, labyrinths, the pyramids. The ancients also knew the lore of numbers and constructed amazing magic squares where the numbers in every row, column and diagonals added up to the same sum.

The famous Indian square, a numerical magic square where every row and groups of related four squares add up to the same number.

This magic square was immortalized by the German painter, Albrecht Dürer, in his painting, "Melancholia," where it's part of the scene's background. What is especially noteworthy is that the painting was done in the year 1514, and the two numbers, 15 and 14, occupy the center of the bottom row of the square. This square is believed to have originated in India. It is the most unusual of the 880 possible solutions to the 4x4 magic square, where not just rows but every symmetrically complementary group of four squares adds to 34.

In fact, magic and puzzles are closely related. Magic tricks (or effects, as the magicians prefer to call them) are puzzles that the audience is expected not to solve, but merely to enjoy for the illusion, for its mystification. A greater puzzle is to create those effects in the first place.

It is not a coincidence that the world's most avid puzzle collectors also have a keen interest in magic, and a more than casual acquaintance with mathematics. Martin Gardner, the beloved author of many books on mathematical games and formerly columnist for Scientific American, is a denizen of all three realms.

Early forms of manipulative or mechanical puzzles include puzzle boxes with sliding panels and secret compartments, puzzle rings (reputedly used as wedding rings to assure fidelity), and a large number of disentanglement variations. Puzzle historian Jerry Slocum has published two books on the types and origins of mechanical puzzles. If you're interested, email me for a source of the books.

Picture puzzles, both flat and as cubes, started out as children's toys. One of my favorite childhood toys was a set of 16 cubes to be arranged as a 4x4 square. Each of the six faces of each cube had a portion of a different fairy tale. If you were clever enough to solve one of them, you could just roll rows of cubes and convert the blocks step by step into each of the other story pictures.

Picture puzzles cut into irregular shapes came a little later and had by a century ago become one of the most popular social entertainments. Fancier and more imaginative forms of the jigsaw puzzle have evolved. Currently the rage is the 3D jigsaw puzzle, where you can build castles and cathedrals and even grandfather clocks, and the shaped jigsaw with exquisite wildlife paintings, like those of Steven Michael Gardner (sold by Bits&Pieces), where there are no straight borders to help you get started.

What all these puzzles have in common is that the finished picture can be formed in only one way. They are mainly a search process requiring patience and a quick eye.

The 7 Tangrams tiles make hundreds of fanciful shapes.

Enter the mighty Tangrams set. Of indeterminate age—at least 150 but maybe much older, say, dating back to Pythagoras and his right triangles—the 7 tangrams pieces (credited by some sources to the Chinese) can form countless different designs. The puzzler is usually given a collection of silhouettes to reproduce with the pieces. The standard shape the pieces make is a square (see illustration).

Tangrams consist of two small triangles, one medium triangle (the size of two small ones joined), a parallelogram and square (also equal to two small triangles), and two large triangles, equal to four small ones. Since every piece is based on half squares, the angles are 90 and 45 degrees and the lengths of the sides of each piece are either the side or the hypotenuse of the triangle. In fact, what looks like a square made of the 7 pieces is actually a diamond, with its "side" equal to the length of two hypotenuses.

Such shapes are also much used in quilting patterns and inlaid mosaics. They are the earliest example of a "polyform" set: differently shaped puzzle pieces based on combinations of multiples ("poly") of the same basic building block ("form").

By our definition, a true polyform set has exactly one of every possible shape that can be made of the unit building block that defines the set, in ascending orders from 1 up to as high as you want to go. The number of units in a piece defines its "order."

So tangrams are not a true polyform set because of having duplicate pieces and not including all sizes from 1 through 4. But as a subset and having just 7 manageable pieces, tangrams are the forerunners of what has evolved since. The most popular version of tangrams available today is made by Rex Games under the trademarked name Tangoes.

The unique characteristic of polyform sets is their inexhaustible fund of puzzle possibilities, and with sometimes thousands of solutions for the same figure. Because the pieces "tile"—they can fill up space with no holes—they serve for countless shapes and sizes of puzzle figures and investigations of amazing phenomena. They are an adventure in continuous discoveries. Here's a line-up of the 3 major polyform families, based on the three "regular" polygons that can tile the plane (square, hexagon, triangle).

Polyominoes

The name comes from the word domino, where a domino is made of 2 squares. So three squares joined become a tromino; four squares a tetromino; five squares a pentomino; six squares a hexomino, and so on. Solomon Golomb introduced the pieces and their names in a lecture to the Harvard Math Club in 1952. The tetrominoes are familiar to most people from the computer game Tetris. And the 12 pentominoes have become almost as well-known as tangrams, even making it into Arthur C. Clarke's sci-fi book, Imperial Earth. As early as the 1920s these shapes were being studied in England and many interesting patterns solved, but they were merely called the fives and the sixes. Even earlier, in the mid-1800s, Sam Loyd created a "broken chessboard" puzzle that used checkerboarded "fives" and a 2x2 square. But it is Golomb's work which set polyominoes firmly into puzzle history.

The Polyominoes are groupings of squares joined along their edges. We show configurations of from 1 through 7 squares, arranged here in 3 symmetrical trays.

Here's a nice little arrangement I discovered that contains all the polyominoes from size 1 through 5.. ...and a companion square with central window, containing all the hexominoes.

The 108 heptominoes in a spectacular solution by David Klarner.

Polyhexes

Polyhexes 1 through 5 in a hexagon-shaped array with center hole (a "hexnut").

Another polygon that can tile with no space left over is the regular hexagon. It is a special puzzle challenge to coax these sets into a symmetrical shape. Some 20 years ago Stewart Coffin produced a set of the size 3 and 4 only, in a nice brick-like hydrastone, and called it the Snowflake puzzle because of its six-fold symmetries.

For a short time, Stewart's Snowflake was being made by Binary Arts in die-cut foam.

A miniature molded plastic set of polyhexes in a see-through box was produced by Tenyo of Japan.

Kadon's sets, sizes 1 through 6, are made of lasercut acrylic. At size 6 one piece has an enclosed hole and so cannot be filled (see illustration below).

Much larger sets are not practical for a hands-on puzzle, but mathematicians and serious puzzlists find large sets a fertile playground for exploration. Computer programs for solving them continue to improve.

The order-6 polyhexes in a hexnut pattern by Michael Keller.

Polyiamonds

This smaller ring contains polyiamonds 1 through 7.

The third regular polygon is the equilateral triangle, and two of them joined make a diamond. Hence three joined are the triamond; four joined make 3 tetriamonds; five joined are the 4 pentiamonds; and there are 12 hexiamonds and 24 heptiamonds. They are shown here in a hexagonal ring pattern I found for sizes 1 through 7.

The 66 octiamonds can form a large hex ring in their own right. The stunning pattern below is by Andrew Clarke.

An interesting fact: the central opening of this pattern can be filled exactly with the 12 hexiamonds. Computer studies have found there are only 55 solutions for this hexiamond center. But the octiamonds have solutions beyond counting. A most remarkable one, by Michael Dowle, divides the 66 octiamonds into 6 congruent triangles that form a large hexagon. Ed Pegg, Jr., shows other interesting constructions at his website, mathpuzzle.com.

This larger ring is all the polyiamonds of order 8.

These three groups are the fundamental polyform families, being derived from the three regular polygons. There are several others more complex, based on right triangles (polytans), rhombs (polyrhombs), circles and arcs, and combination tilings such as octagons and squares, or hexagon/square/triangle tessellations, that I'll describe in the future. These are truly playable art because their arrangements plot into mosaic patterns and quilt designs that have since antiquity been part of the human synthesis of mathematics and aesthetics... in short, one of the ways humans have fun while creating beauty out of structure, order out of chaos.

- Kate Jones

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