Suppose you are given two polygons with equal areas. Can you cut either of them into a finite number of smaller polygons that can be rearranged to form the other? It turns out that this is always possible: the Wallace–Bolyai–Gerwien theorem shows that any two such polygons can both be cut into pieces that can be arranged into a rectangle whose shape depends only on the area in question. By further subdiving that rectangle into pieces that are the intersections of pieces from both sets, a single set of pieces can be found that can be rearranged to form either of the original polygons.
In more detail:
We won’t spell out all the fine points needed to turn this into a completely general algorithm, but we will illustrate the result of cranking the handle and going through these steps for a pair of polygons with no special symmetries.
[More to come soon ...]