Ladies' Diary/Dutchmen's Three Wives/Proof 1

Proof
Let the number of hogs bought by any one of the women be $x$.

Let the number of hogs bought by her husband be $x + n$.

Then:
 * the money paid by the woman is $x^2$ shillings
 * the money paid by her husband is $\paren {x + n}^2$ shillings

There are $21$ shillings to the guinea, so:


 * $\paren {x + n}^2 = x^2 + 63$

and so:
 * $x = \dfrac {63 - n^2} {2 n}$

We see that in order for $x$ to be an integer it is necessary for $n$ to be odd.

Now we can plug various values of $n$ and see where this gets us.

It turns out by trying various $n$ that $3$ such values work, as follows:

If $n = 1$ then $x = 31$ and $x + n = 32$

If $n = 3$ then $x = 9$ and $x + n = 12$

If $n = 7$ then $x = 1$ and $x + n = 8$.

Now we can see that:
 * Hendrick bought $32$ hogs and Catriin bought $9$ hogs
 * Claas bought $12$ hogs and Geertrick bought $1$ hog

and the allocation of men with their wives then becomes apparent.