Harold's army: sixty-one squares, then one giant square
An old chronicle claims Harold's Saxon army at Hastings stood in 61 equal square formations, the same whole number of men on every side of each. When Harold himself joined the fight, the entire army re-formed into one single giant square. Henry Dudeney's puzzle: is this arithmetically possible, and if so, what's the smallest army size for which it works?
Reveal the answer
Yes, but only with an absurdly huge army: 3,119,882,982,860,264,400 men (over 3 quintillion), with 51,145,622,669,840,400 men in each of the 61 squares, plus Harold forming one giant square 1,766,319,049 men on a side. It's a case of the Pell equation 61x²+1=y², which Fermat once posed as a challenge to English mathematicians. From Henry Dudeney's Amusements in Mathematics (1917), puzzle No. 129, "The Battle of Hastings."
— Henry Dudeney, Amusements in Mathematics — Puzzle No. 129, "The Battle of Hastings," 1917