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Tangrams are a well-known dissection of the square into seven convex polygons. One fun mathematical question is: how many convex rearrangements of the seven pieces are there?

Answer: there are 12 more. This is a theorem of Fu Traing Wang and Chuan-Chih Hsiung from 1942.

The Stomachion is a dissection of the square into fourteen pieces, apparently studied by Archimedes. In how many ways can these pieces be reassembled into a convex polygon?

There are at least two versions of this: how many different convex shapes, and then how many rearrangements of the pieces for each shape. (People think that Archimedes studied this second question for the square.)

I'm also interested in the same questions for the Stomach, a closely related 11-piece dissection.

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    $\begingroup$ The "Sei Shonagon Chie no Ita" is another dissection puzzle where the question of the number of distinct convex rearrangements has been studied arxiv.org/abs/1407.1923 . The solution depends on Wang and Hsiung's techniques, which appear to be special to dissections made of pieces which can all be decomposed into congruent isoceles right triangles. The Stomach(ion) doesn't appear to have this property. $\endgroup$ – j.c. Apr 23 '16 at 19:36
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    $\begingroup$ Is there an algorithm that can answer these questions, even if the algorithm is uselessly slow? $\endgroup$ – Joseph O'Rourke Apr 23 '16 at 19:50
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The Logelium page of Bernd Karl Rennhak claims that there are 637 convex polygons that can be formed. There are linked pages with pictures of the different types of polygons and the numbers of solutions.

edit: This is not a complete answer to your question. The ones depicted in the pages linked to above are built from the "simplified" tiles of the Stomach puzzle and furthermore satisfy the restriction that the tiles "conform to the base grid". I couldn't find a precise definition but this is described roughly at the bottom of this page.

It seems that Rennhak found these solutions via his software which is capable of finding solutions to tiling problems that fit on an integer grid. See these pages for some more details.

In any case it would probably be worth writing him if you are interested in more of the missing details.

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