Hilbert’s Third Problem, presented in 1900, asked: given two polyhedra of equal volume, are they scissors-congruent? That is, can one always be cut into a finite number of pieces and be reassembled into the other? In two dimensions, it is clear that two polygons have equal area if they are scissors congruent, but the converse was proved in 1807. The parallels between flat origami and polygonal decomposition suggest a common framework, which motivates us to define a notion of fold-congruence. We pose the question: are two polygons of equal area always fold-congruent? In this talk, we discuss preliminary investigations into this seemingly difficult question.