Greek Anthology Book XIV: 48. - Problem

Problem

The Graces were carrying baskets of apples, and in each was the same number.

The nine Muses met them and asked them for apples, and they gave the same number to each Muse,

and the nine and three had each of them the same number.

Tell me how many they gave and how they all had the same number.

Solution

According to tradition, there are three Graces.

Let $n$ be the number of apples owned by each party at the end of the transaction.

At the end of the transaction, there are $12$ parties each with the same number $n$.

Each of the three Graces then had $4 n$ apples at the start.

In total there were $12 n$ apples.

Any (strictly) positive integer $n$ satisfies the conditions.

$\blacksquare$

Sources

• 1918: W.R. Paton: The Greek Anthology Book XIV ... (previous) ... (next): $48$. -- Problem