Essential Polyforms — Addendum

The Tan TricksTM series

One of the more complex of the polyform genre is the series of shapes made by joining isosceles right triangles along their matching edges in every possible combination. The family name of this set is polytans, borrowing the word tan from the classic Tangram set. Sometimes mathematicians also refer to polytans as polyaboloes.

From chapter 11, "Polyhexes and Polyaboloes," of Martin Gardner's Mathematical Magic Show:

This family of pieces was first discussed in print by Thomas H. O'Beirne in New Scientist, December 21, 1961, in a column on recreational mathematics that he then contributed regularly to the magazine. The pieces had been suggested to him by S. J. Collins of Bristol, England, who gave the name "tetraboloes" to the order-4 set because the diabolo, a juggling toy, has two isosceles right triangles in its cross section. This implies the generic name "polyaboloes."

While the word "diabolo" has some dictionary associations with "devil," in this case the word is rooted in the Greek for "across" (dia-, as in diameter or diagonal) and "throwing" (ballo). Thus it has more kinship with the gauchos' balo or bolo than with anything diabolical. Calling the puzzle set polytans eliminates the confusion.

Since each unit triangle is half a square (a square cut in half diagonally), the pieces always want to reunite the two halves to form squares again. It takes a little practice to recognize the tile edge that is the hypotenuse, which must be oriented diagonally for the tile to fit properly within its grid.

We offer the series from 1 through 6 in three separate, progressively larger square trays. The smallest contains sizes 1, 2 and 4. The middle size holds sizes 3 and 5. The largest has all the tiles of size 6. Each size is given its own color for easy sorting. We use beautifully tinted transparent acrylic, precision lasercut. The sizes are, of course, all compatible for mixing and matching in larger figures. A large workbook includes challenges for the whole series.

The various sizes have their own names as well, with the prefix indicating how many triangles the piece contains:   unit, ditan, tritan, tetratan, pentatan and hexatan.

That the sizes 1 through 5 would divide so neatly into two squares was discovered by Michael Keller of Baltimore.


Polytans of sizes 1, 2, and 4 fill a square 5" tray. Each size has its own color. There is an extra piece of the unit size triangle, 3 ditans and 14 tetratans. The tray is a deceptive shape. It is not truly a square but a diamond. The grid is diagonal within the frame. Tan Tricks I and Tan Tricks II, below, are one integral set and are sold as a combo, although packaged and playable individually. A 5-piece subset of the tetratans presents a special activity of finding symmetrical combinations. Tan Tricks I is recommended for ages 10 to adult. Value alone, $29

The 5-piece subset of tetratans explored by Tick Wang has now expanded to quartets, 20 pieces in a tray, each group of five pieces in a different color. Available by custom order. See the Tetra-Tray collection, where you can also order it.

The 4 tritans and 30 pentatans fill an 8" square tray. In this case the grid lies squarely within the tray, and the hypotenuse in any piece will plot diagonally. It is no mean feat to solve it so that the tritans form a connected group. This set is sold as a combo with Tan Tricks I, although packaged and playable separately. For persistent puzzlers, ages 12 to adult. Value alone, $65

The combo, both sets together, color-coordinated so all 5 sizes have their own color, $94

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The grand array of 107 hexatans has an extra plug made of 6 singles that can duplicate the shape of any hexatan. We show them making the symmetrical "house" at the very center (solution by Anneke Treep of the Netherlands). They could form other shapes as well or be split up and placed anywhere within the tray. The "house" is the most compact symmetrical plug that allows the whole tray to be filled. With it at the center took almost 40 hours to solve by hand. An even more symmetrical hexatan, the stretched hexagon, has been proven to make it impossible to fill the whole tray when it is the duplicate. See also a remarkable triangular shape solved by Nick Maeder. The 16x16" tray comes with a clear lid. For hardiest solvers only. $195

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