# How many convex shapes can be made with the pieces of the Stomachion?

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?

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.

• 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.
– j.c.
Apr 23, 2016 at 19:36
• Is there an algorithm that can answer these questions, even if the algorithm is uselessly slow? Apr 23, 2016 at 19:50