Alcuin's four-sided city: how many houses fit inside, from the side lengths alone?
In one of the 53 brain-teasers an 8th-century scholar wrote to sharpen young minds, a walled city has four straight sides measuring 1,100, 1,000, 600 and 600 feet. Each house inside measures 40 by 30 feet. Using nothing but the side lengths, how many houses can the city hold?
Reveal the answer
Alcuin's own answer was 520, found by averaging the two pairs of opposite sides (1,050 and 600) and multiplying — a method that only works for rectangles, not this trapezoid-shaped city. The true area gives roughly 523 house-plots, and later mathematicians trying to physically pack 40-by-30 rectangles into the real shape have only managed around 510 to 517. Even the 'answer' in the oldest known collection of math puzzles in Latin doesn't quite hold up.
— Alcuin of York, Propositiones ad Acuendos Juvenes — Problem 27, De Civitate Quadrangula, c. 800 AD