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.