cannonball stacking | cannonball stacking problem

Find a way to stack a square of cannonballs laid out on the ground into a square pyramid (i.e., find a square number which is also square pyramidal). This corresponds to solving the Diophantine equation sum_(i = 1)^k i^2 = 1/6 k(1 + k)(1 + 2k) = N^2 for some pyramid height k. The only solutions are (k, N) = (1, 1) and (24, 70) (Ball and Coxeter 1987, Dickson 2005), as conjectured by Lucas, partially proved by Moret-Blanc and Lucas, and proved by Watson. Watson's proof was almost elementary, disposing of most cases by elementary means, but resorting to the use of elliptic functions for one pesky case. Entirely elementary proofs have been given by Ma and Anglin.

sphere packing | square number | square pyramid | square pyramidal number