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The Tangramion

Put-Together Puzzles

by Serhiy Grabarchuk

Click this image to go to the Printable Puzzle page.Printable Pieces

The Tangramion.
This is a new assembling puzzle consisting of seven two-sided tiles which initially form a perfect square as that shown at left. Using all the seven tiles, you can solve different challenges both entertaining and cognitive. Some of them are posed below:

   1. Form different shapes shown below with all seven tiles each time*.
   2. What's the ratio of all the seven tiles of the Tangramion?
   3. A square can be formed in several different ways. Can you find all of them?
   4. What's a total number of convex shapes which can be formed, using all the seven tiles each time?

Note, that tiles can be rotated and flipped over, but not overlapped.

Assemble these shapes, using all the seven tiles of the Tangramion.
The Tangramion Background
This new assembling or put-together puzzle has been devised as a result of a thorough analysis of two classic puzzles -- Archimedes' Square Stomachion2 and the T. A. Snider Diamond Puzzle.

Archimedes' Square Stomachion.

Diagram of the T. A. Snider Diamond Puzzle.
The first puzzle has a long history, and is, in fact, the oldest known puzzle of this type, and, perhaps, the oldest puzzle ever confirmed in a written document. It's attributed to Archimedes, and has more than 22-century history. Its ancient origin has been proved after amazing modern investigations of the Archimedes Palimpsest, a 174-page volume, created a thousand years ago, then lost and found and again lost, until it was finally rediscovered in the last decade of the 20th century. Then it was carefully restored and exactly translated. One of the Archimedes Palimpsest's parts describes The Stomachion or Loculus of Archimedes (Archimedes' Box).3

Archimedes' puzzle contains 14 different tiles, and it might seem that such a puzzle set proposes very wide and varied possibilities for creative puzzle play. But, in fact, numerous and mostly inconvenient tiles make it somewhat complicated creating and solving challenges and having natural fun with this very old and famous puzzle. And this is what makes Archimedes' Square Stomachion rather a math object than a really fun puzzle.

There were some attempts to reduce the total number of tiles so that to add more fun to the puzzle, and at the same time not to lost its original taste and save as much as possible of its properties. One of these attempts was resulted with 11 tiles, again giving very nice math puzzle, but still not too much adding puzzling fun to it.

One day, comparing Archimedes' Square Stomachion (a) and the T. A. Snyder Diamond Puzzle (b) I've discovered that surprisingly many dividing lines of their grids exactly match, and pro tanto many tiles are exactly the same.
Compared diagrams: From Archimedes' Stomachion and the Diamond Puzzle to the Tangramion.
Moreover, if in the traditional drawing of Archimedes' Square Stomachion we exchange two blocks of tiles as shown (c), we can easily spot even more of the same elements in the puzzles what is very thrilling if to take into consideration the fact that both of them use a very unusual pattern of their grids. In particular, the Diamond Puzzle has a rare grid with the combination of its diagonals running at 45 and 26,565... degrees almost not used in other Tangram-like puzzles. And the very same degrees for main diagonals of its grid are used in Archimedes' Square Stomachion.

In the third diagram (c) the main dividing lines (shown in red) isolate six clearly visible regions exactly coinciding with regions of the Diamond Puzzle and containing from one to four original tiles of Archimedes' Square Stomachion. Now it's pretty easy to spot the most natural way how to merge some of the original tiles so that to reach the goal of reducing the number of pieces and making them more convenient, and as a result, to make the puzzle more playable and fun.

Additionally, having 14 tiles in Archimedes' Square Stomachion and keeping in mind "standard" 7 tiles in the traditional Tangram, the very pleasant possibility of getting the very same number of tiles in our new puzzle was quite appealing. And so, adding a short vertical line to the whole red pattern (c), finally we have our set of tiles (d). By the way, you can notice that the tiles form the 7 shape in the middle of the square (d).

Lastly, combining two puzzle names, Stomachion and Tangram, we got the name for this new puzzle, Tangramion.

Printable Pieces    

Printable Puzzle: To solve the challenges of the Tangramion you can print its tiles. For this, click the image marked with the "pp" pictogram (as that shown at left) to go to a new window with the puzzle tiles; then you can print them and cut them out.

PDF Version    

PDF Version: Also, you can download a PDF file of the puzzle, and then print it out if you wish. For this, use the link marked with the "pdf" pictogram (as that shown at left).

Copyright Note

Copyright Note: Note that the entire puzzle and its challenges are copyrighted, so you can print them for your own use only, and not for any kind of commercial profit.
Notes & References
     *) It is the first series of shapes; the second series will follow. Shapes marked with (~) were created by Tanya Grabarchuk. Shapes marked with (*) were created by Helen Homa. The remaining shapes are created by myself.
     2) Because of slightly different translations of the Archimedes Palimpsest's original texts there were described two different basic versions of Archimedes' Stomachion,
square and rectangular. The latter is nothing other than a square version stretched to a 1 x 2 rectangle. The square version is "more" traditional, and so it's used in my researches.
     3) Gianni A. Sarcone and Marie-Jo Waeber at their
Archimedes' Lab website propose more reasonable and appropriate naming for Archimedes' Stomachion.
     4) At the
Kadon Enterprises, Inc. website you can find their wonderful, hands-on Archimedes' Square puzzle with all, recently discovered 536 solutions for a square shape, plus the 637 other convex shapes.
     5) In Spring of 2012,
Kadon Enterprises, Inc. released a wonderful, laser-cut acrylic version of the Tangramion accompanied with a 16-page booklet. The set of challenges is extended by Kate Jones so that the whole set proposes over 200 different challenges.
Last Updated: April 6, 2012
Posted: October 26, 2007
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