Puzzles
Edouard Lucas's cannonball problem
Stack cannonballs into a square pyramid: 1 ball on top, 4 beneath that, 9 beneath that, and so on down to a square base N balls per side. Is there any pyramid height for which that exact same pile of cannonballs could instead be laid out as one flat square array, with none left over and none missing?
Reveal the answer
Yes, but remarkably only once: a 24-layer pyramid holds exactly 4,900 cannonballs, and 4,900 = 70 squared, a perfect 70-by-70 square. Edouard Lucas posed this as a number-theory challenge in the 1870s; it took until 1918 for G. N. Watson to actually prove no other pyramid height works (besides the trivial single ball), using elliptic functions.
— Edouard Lucas, Cannonball problem — posed 1870s; proved by G. N. Watson, 1918