Greek Anthology Book XIV: Metrodorus: 119

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Arithmetical Epigram of Metrodorus

Ino and Semele once divided apples among twelve girl friends who begged for them.
Semele gave them each an even number and her sister an odd number,
but the latter had more apples.
Ino gave to three of her friends three-sevenths,
and to two of them one-fifth of the whole number.
Astynome took eleven away from her
and left her only two apples to take to the sisters.
Semele gave two quarters of the apples to four girls,
and to the fifth one sixth part,
to Eurychore she made a gift of four;
she remained herself rejoicing in the possession of the other four apples.


Solution

The wording of the epigram is ambiguous and unclear.

It is assumed that Ino and Semele are sisters, and the sentence:

Semele gave them each an even number and her sister an odd number

means:

Semele gave them each an even number and her sister (Ino) (gave them each) an odd number


Let $i$ be the number of apples that Ino distributed.

Let $s$ be the number of apples that Semele distributed.


We interpret the epigram as stating that:

$\dfrac 3 7$ of Ino's apples were given to be shared between $3$ of her friends
$\dfrac 1 5$ of Ino's apples were given to be shared between $2$ of her friends
$11$ of Ino's apples went to Astynome

and:

$2$ of Ino's apples remained to Ino and Semele.


$\dfrac 2 4$ of Semele's apples were shared between $4$ of her friends
$\dfrac 1 6$ of Semele's apples were given to a $5$th friend
$4$ of Semele's apples were given to Eurychore.
$4$ of Semele's apples were kept by Semele.


First we investigate Ino's share:

\(\ds i\) \(=\) \(\ds \dfrac {3 i} 7 + \dfrac i 5 + 11 + 2\)
\(\ds \leadsto \ \ \) \(\ds 35 i\) \(=\) \(\ds 15 i + 7 i + 35 \times 13\) multiplying through by $35 = \lcm \set {5, 7}$ and simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {35 - 15 - 7} i\) \(=\) \(\ds 35 \times 13\)
\(\ds \leadsto \ \ \) \(\ds 13 i\) \(=\) \(\ds 35 \times 13\)
\(\ds \leadsto \ \ \) \(\ds i\) \(=\) \(\ds 35\)


So Ino started with $35$ apples, of which:

$15$ of Ino's apples were given to be shared between $3$ of her friends
$7$ of Ino's apples were given to be shared between $2$ of her friends

and as we know:

$11$ of Ino's apples went to Astynome

and $2$ of Ino's apples remained to Ino and Semele.


Then we investigate Semele's share:

\(\ds s\) \(=\) \(\ds \dfrac {2 s} 4 + \dfrac s 6 + 4 + 4\) and we note that $\dfrac 2 4$ is not expressed in canonical form
\(\ds \leadsto \ \ \) \(\ds 12 s\) \(=\) \(\ds 6 s + 2 s + 12 \times 8\) multiplying through by $12 = \lcm \set {4, 6}$ and simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {12 - 6 - 2} s\) \(=\) \(\ds 12 \times 8\)
\(\ds \leadsto \ \ \) \(\ds 4 s\) \(=\) \(\ds 12 \times 8\)
\(\ds \leadsto \ \ \) \(\ds s\) \(=\) \(\ds 12 \times 2\)
\(\ds \) \(=\) \(\ds 24\)


So Semele started with $24$ apples, of which:

$12$ of Semele's apples were shared between $4$ of her friends
$4$ of Semele's apples were given to a $5$th friend

and as we know:

$4$ of Semele's apples were given to Eurychore.

and $4$ of Semele's apples were kept by Semele.

$\blacksquare$


The fact that Semele gave them an even number and Ino gave them an odd number is unnecessary information, and in fact misleadingly inaccurate.

In particular, if Ino gave $7$ apples to be shared between $2$ friends, one of them definitely received an even number of apples.


Source of Name

This entry was named for Metrodorus.


Sources

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