Triggered by the MO question, "How many convex shapes can be made with the pieces of the Stomachion?How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question:
Q. Given $n$ polygons in a set $S$, say each with integer coordinate vertices, is there some algorithm to count/list the number of ways that some subset of $S$ of these polygons could be fit together, with pairwise disjoint interiors, such that their union is a convex polygon?
The polygons may be rotated and translated in the fitting-together. One might restrict the polygons in $S$ each to be convex. Specifying integer coordinates is meant to make the input to the question of finite length, say, $L$ bits.
Examples from the page that j.c. found:
[![StomachParts][1]][1]
(Image from [Bernd Karl Rennhak](http://www.logelium.de/Stomachion/StomachionPuzzel_EN.htm).)
The difficulty is that it is not immediately evident how to reduce the problem to a finite set of possibilities to try, for the gluing-together need not be whole-edge to whole-edge. Maybe decidability of 1st-order theories of reals can be applied?