Given any two bounded subsets A and B of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of A and B into a finite number of disjoint subsets, A = A_1 ∪ ... ∪ A_k, B = B_1 ∪ ... ∪ B_k, such that for each i between 1 and k, the sets A_i and B_i are congruent.