Proof. By construction, if b appears immediately below a in the array, then their
column number c is given by c = (b−a) mod k. This means that if (b−a) mod k =
k − 1, then d1 and d2 both have column number 0. Otherwise, d1 and d2 both have
column number ((b − a) mod k) + 1.
We continue by analyzing the row numbers of these fillings.