Kadon Enterprises, Tan Tricks I, tetratan symmetries

Symmetrical solutions with 5 tetratans

Symmetries, the full collection
Artistic figures
Grand quartets

This project was initiated by Tick Wang, a puzzle enthusiast from Shanghai, in January 2022, as an addendum to our Tan Tricks sets. If you already own the set, use the pieces shown and have fun solving. Even if you don't own the set, you can just make cardboard cutouts of these five shapes.

Tick examined the 14 tetratans of Tan Tricks I and set aside all the pieces that were convex or symmetrical. He was left with these five rather irregular tiles:

The 5-piece subset of tetratans chosen by Tick Wang

Tick's goal was to find all the symmetrical shapes that could be formed by 2, 3, 4, and all 5 of these tetratans. He noticed that certain two pieces always occurred, in every shape, and the question is: Can any 3- or 4-piece symmetrical shape be formed without including both of those pieces? Can you find a proof either way? The first solver to send us such a proof will win a prize. Our email address: Kadon Enterprises, Inc.

Here are a few sample solutions using 2, 3, 4, and all 5 pieces. Do you detect which two pieces are always present? Piece colors are shown as different for artistic effect only, to highlight each piece.

A few sample symmetries by Kate Jones

The full collection
Here is the entire collection of the 160 symmetrical shapes derived by George Sicherman that can be solved with Tick's quaint quintet. It is interesting to find that there are three kinds of symmetry:

Orthogonal (mirror symmetry vertical on the grid): 38

Diagonal (tilted by 45 degrees on the grid): 64

Rotational (turned halfway, like the letters S and Z): 58

Further, there are 55 figures with holes enclosed, anywhere from a single triangle to 6 triangles.

Have fun solving these. Also see how many 3-piece and 4-piece symmetries you can find.

George Sicherman's computer-derived collection of all symmetrical shapes with 5-piece tetratan subset

Now to improvise artistic shapes...
Another way to play with this handful of tricky pieces is to invent and build funny figures. Tick Wang is famous for his imaginative free-form characters. Here are a couple of cute shapes by Kate to amuse him: a goose and a puppy.

The 5-piece subset of tetratans can improvise art

Tetra-Tray—Tick's grand quartets
In August 2022, Tick's creativity went one step farther, with "Tetra-Tray" combinations of four sets of his five tetratan pieces forming kaleidoscopic symmetries in a square tray. Each 5-piece set has its own color, and Tick added one hole in each piece to enhance the artistic design of the assembled patterns and get a quick count of how many pieces were used.

Tetra-Tray

Here we show his huge collection of mirror and rotational symmetries for your visual pleasure. They start with 2 pieces of each color, then 3, 4, and all 5. See not only the interesting shapes the tiles will make but also the variety of spaces around them, including triangles, squares, parallelograms, trapezoids, stars, and even octagons. Isn't it amazing how many variations just 5 shapes can make, and even their symmetrical solutions can be rotated or exchanged to form even more permutations of a simple arrangement? Here is the wonder of mathematics formed into playable art. Enjoy the tour.

Tetra-Tray symmetries with 2 and 3 tetratans

To order your own set of Tetra-Tray in lasercut acrylic (the 20 tetratan pieces in 6" tray) to work out these solutions or to discover your own designs and entertain your family and friends, click on the green logo. $29

Back to Tan Tricks I