Greek Anthology Book XIV: 3. - Problem

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:

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