Greek Anthology Book XIV: 3. - Problem

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Problem

Cypris thus addressed Love, who was looking downcast:
"How, my child, has sorrow fallen on thee?"
And he answered:
"The Muses stole and divided among themselves, in different proportions, the apples I was bringing from Helicon, snatching them from my bosom.
Clio got the fifth part,
and Euterpe the twelfth,
but divine Thalia the eighth.
Melpomene carried off the twentieth part,
and Terpsichore the fourth,
and Erato the seventh;
Polyhymnia robbed me of thirty apples,
and Urania of a hundred and twenty,
and Calliope went off with a load of three hundred apples.
So I come to thee with lighter hands, bringing these fifty apples that the goddesses left me."


Solution

Let $n$ be the number of apples Love started out with.

Then:

$\dfrac n 5$ were stolen by Clio
$\dfrac n {12}$ were stolen by Euterpe
$\dfrac n 8$ were stolen by Thalia
$\dfrac n {20}$ were stolen by Melpomene
$\dfrac n 4$ were stolen by Terpsichore
$\dfrac n 7$ were stolen by Erato
$30$ were stolen by Polyhymnia
$120$ were stolen by Urania
$300$ were stolen by Calliope
$50$ were salvaged by Love.


Thus:

\(\ds n\) \(=\) \(\ds \dfrac n 5 + \dfrac n {12} + \dfrac n 8 + \dfrac n {20} + \dfrac n 4 + \dfrac n 7 + 30 + 120 + 300 + 50\)
\(\ds \leadsto \ \ \) \(\ds 840 n\) \(=\) \(\ds 168 n + 70 n + 105 n + 42 n + 210 n + 120 n + 840 \times 500\) multiplying through by $840$
\(\ds \leadsto \ \ \) \(\ds 125 n\) \(=\) \(\ds 840 \times 500\) gathering terms
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 840 \times 4\) dividing both sides by $125$
\(\ds \) \(=\) \(\ds 3360\)


Thus

$672$ apples were stolen by Clea
$280$ apples were stolen by Euterpe
$420$ apples were stolen by Thalia
$168$ apples were stolen by Melpomene
$840$ apples were stolen by Terpsichore
$480$ apples were stolen by Erato


in addition to:

the $30$ that were stolen by Polyhymnia
the $120$ that were stolen by Urania
the $300$ that were stolen by Calliope
the $50$ that were salvaged by Love


and it is seen that:

$672 + 280 + 420 + 168 + 840 + 480 + 30 + 120 + 300 + 50 = 3360$

$\blacksquare$


Sources

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