Let be a positive integer. Let be a subset of with size . We denote the complement of with respect to as .

Let be an element of .

Let be the number of ordered pairs such that .

Theorem. For all , .

Proof. Let . Let . They are the same size: . So their differences and are the same size too. But is precisely and is precisely (why?).

Remark. This is more or less what's known as the "Hexachordal theorem" in "musical set theory". The usual interpretation is when , is identified with the pitch classes in twelve-tone equal temperament, and is a "hexachord", a set of six pitch classes. Apparently there is also a crystallography interpretation.

This is a paper on a generalization and I love how many citations they managed to come up with for proofs, some of which are apparently more concise than others: http://cs.smith.edu/~jorourke/Papers/CHexa-MCM.pdf