Dissection problems tend to be pretty hard; for instance, to my knowledge we don't know whether it's possible to dissect an equilateral triangle into $k$ disjoint congruent regions for any $k$ whose squarefree part is not 2, 3, 5, or 6.
As the example for $k=5$ shows, questions of congruent dissection like this are not at all obvious, so assertions that a given dissection is impossible are perhaps hard to trust without rigorous proof.
However, I don't have a good sense of what techniques are available for such questions. Are we able to easily characterize, for instance, those shapes which have a dissection into two congruent regions? Are there proof methods that yield at least partial negative results in some cases? How tractable are problems like this, in general? Even a pointer to a single impossibility result for congruent dissections (perhaps under some restrictions, e.g. polygonal regions) would be welcome.
