More on tangrams
If we cut a unit square in half, we get two triangular pieces. These pieces are the building blocks of polyabolos, a type of polyanimal. Given 16 polyabolo pieces, 20 convex shapes can be formed, by gluing the 16 pieces together in different ways.
Notice that each of the 7 pieces of the tangram, seen below, is a polyabolo. The tangram then consists of 16 of the small triangles that are found in polyabolos. So a natural question arises. How many of the convex polygons, that can be made from 16 polyabolo triangles, can be made using the 7 pieces of the tangram? In 1942, Fu Traing Wang and Chuan-Chih Hsiung showed that the answer is 13.
The tangram is an old and popular puzzle, and so it has several variants which were popular or interesting enough to be named. Here are the ones I'm aware of, along with the number of convex polygons that can be formed by rearranging their pieces:
For lots more interesting results, read this. They also offer a document in French, which I have translated using Google Translate. You can download that here. If you are at all interested in the properties of tangrams, you will surely want it.