212. Divide one hundred measures of wheat among one hundred persons so that each man receives 3, each woman 2, and each child $\frac{1}{2}$ measure. How many men, women, and children?
212. We have the system: $3 x+2 y+\frac{1}{2} z=100$ and $x+y+z=100$. Eliminating $2:$

$$
200-6 x-4 y=100-x-y ; 100=5 x+3 y
$$

or $20=x+\frac{3 y}{5}$. Since $y$ is an integer, it must be a multiple of 5, for example $5 u$, then $y=5 u$ and $20=x+3 u$, from which $x=20-3 u$. It is clear that $u$ can take the values $0,1,2$, $3,4,5,6$. Accordingly, we get 7 solutions:

Men $x=20,17,14,11,8,5,2$.

Women $y=0,5,10,15,20,25,30$.

Children $\quad \boldsymbol{z}=80,78,76,74,72,70,68$.

Alcuin gives only one solution: $11,15,74$.