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

Puzzle

There came $3$ dutchmen of my acquaintance to see me, being lately married;

they brought their wives with them.

The men's names were Hendrick, Claas, and Cornelius;

the women's Geertrick, Catriin, and Anna;

but I forget the name of each man's wife.

They told me that they had been at market, to buy hogs;

each person bought as many hogs as they gave shillings for each hog;

Hendrick bought $23$ hogs more than Catriin,

and Claas bought $11$ more than Geertrick;

likewise, each man laid out $3$ guineas more than his wife.

I desire to know the name of each man's wife.

Solution

Note that a guinea is $21$ shillings.

Hendrick bought $32$ hogs and his wife Anna bought $31$

Claas bought $12$ hogs and his wife Catriin bought $9$

Cornelius bought $8$ hogs and his wife Geertrick bought $1$.

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.

$\blacksquare$

Historical Note

This proof of the problem of the Dutchmen and their $3$ wives is reported as being the work of J. Hill, who may well have been the same John Hill who published Arithmetick, Both in the Theory and Practice in $1716$.

Sources

• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Ladies' Diary or Woman's Almanac, $\text {1704}$ – $\text {1841}$: $141$