Seventeen Horses

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Classic Problem

A man dies, leaving $17$ horses to be divided among his heirs.

They are to be distributed in the ratio $\dfrac 1 2 : \dfrac 1 3 : \dfrac 1 9$.

How can this be done?


General Problem: $1$

A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $3$ heirs.

They are to be distributed in the ratio $\dfrac 1 a : \dfrac 1 b : \dfrac 1 c$.

Let $\dfrac 1 a + \dfrac 1 b + \dfrac 1 c < 1$.

Then there are $7$ possible values of $\tuple {n, a, b, c}$ such that the required shares are:

$\dfrac {n + 1} a, \dfrac {n + 1} b, \dfrac {n + 1} c$

These values are:

$\tuple {7, 2, 4, 8}, \tuple {11, 2, 4, 6}, \tuple {11, 2, 3, 12}, \tuple {17, 2, 3, 9}, \tuple {19, 2, 4, 5}, \tuple {23, 2, 3, 8}, \tuple {41, 2, 3, 7}$

leading to shares, respectively, of:

$\tuple {4, 2, 1}, \tuple {6, 3, 2}, \tuple {6, 4, 1}, \tuple {9, 6, 2}, \tuple {10, 5, 4}, \tuple {12, 8, 3}, \tuple {21, 14, 6}$


General Problem: $2$

A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs.

They are to be distributed in the ratio $\dfrac 1 {a_1} : \dfrac 1 {a_2} : \cdots : \dfrac 1 {a_m}$.

Let $t = \dfrac q r = \ds \sum_{k \mathop = 1}^m \dfrac 1 {a_k}$ expressed in canonical form.

Let $t \ne 1$.

Then it is possible to achieve the required share by adding $s$ objects to the existing $n$ such that:

$s + q = r$

when $q = n$.


This still works whether $q$ is positive or negative.


Solution

One son gets $9$ horses, another gets $6$, and the final son gets $2$.


Proof

We note that:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 9 = \dfrac 9 {18} + \dfrac 6 {18} + \dfrac 2 {18} = \dfrac {17} {18}$

The fact that the ratios do not add up to $1$ tells us straight away that those ratios are not in fact the actual fractions of the inheritance that each will receive.

It is noted that the question carefully does not state that one son receives $\dfrac 1 2$, the next son $\dfrac 1 3$ and the third $\dfrac 1 9$, because that is not what actually happens here.


The solution is usually couched in the form of a story, along the following lines:

A stranger passing through on horseback heard the sons squabbling amongst their string of horses, and stops and listens to their argument.
Dismounting, he says to them, "Calm down, gentlemen! Let us see what we have here."
He leaves his horse to go and mingle with the other horses, and says to the youngest son:
"Go ahead and select one ninth of the horses here, but please leave my horse alone."
And the youngest son does so, and puts a halter over the neck of each of two of the finest horses.
The stranger continues to the second son: "Go ahead and select one third of the horses here, but not those that your brother has chosen, and again, please leave my horse alone."
The second son does so, carefully counting, and puts a halter over the neck of each of six of the remaining horses.
The stranger continues to the first son: "The remaining horses are yours, but please leave my horse alone."
And as the first son puts a halter over the neck of each of the nine remaining horses of his father, the stranger remounted his own horse and went on his way.

$\blacksquare$


Historical Note

David Wells tells us in his Curious and Interesting Puzzles of $1992$ that the Seventeen Horses riddle appears in one of Niccolò Fontana Tartaglia's works: either Quesiti, et Inventioni Diverse of $1546$ or General Trattato di Numeri et Misure of $1556$.

Wells, however, does not tell us which one of those two it appears in.


It has since appeared in countless anthologies of puzzles and riddles.


Sources

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