# Alcuin’s Corn Problem

Alcuin of York (735-804) had a series of similar problems involving the distribution of corn among servants.  Since the three propositions were the same format with only the numbers changing, I thought I would present them in a more concise form:

“Proposition

A certain head of household had a number of servants, consisting of men, women, and children, among whom he wished to distribute quantities, modia, of corn.  The men should receive three modia; the women, two; and the children, half a modium.

(a)  If the head of household has 20 servants and wished to distribute 20 modia of corn among them, let him say, he who can, How many men, women and children must there have been.

(b)  If the head of household has 30 servants and wished to distribute 30 modia of corn among them, let him say, he who can, How many men, women and children must there have been.

(c)  If the head of household has 100 servants and wished to distribute 100 modia of corn among them, let him say, he who can, How many men, women and children must there have been.”

I will give Alcuin’s solutions first, followed by my more expansive solutions that rely on our familiar symbolic algebra that was not available in Alcuin’s time.