Greek Anthology Book XIV: Metrodorus: 120

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Arithmetical Epigram of Metrodorus

The walnut-tree was loaded with many nuts, but now someone has suddenly stripped it.
But what does he say?
"Parthenopea had from me the fifth part of the nuts,
to Philinna fell the eighth part,
Aganippe had the fourth,
and Orithyia rejoices in the seventh,
while Eurynome plucked the sixth part of the nuts.
The three Graces divided a hundred and six,
and the Muses got nine times nine from me.
The remaining seven you will find still attached to the farthest branches."


Solution

Let $n$ be the number of walnuts that were originally on the tree.

$\dfrac n 5$ went to Parthenopea
$\dfrac n 8$ went to Philinna
$\dfrac n 4$ went to Aganippe
$\dfrac n 7$ went to Orithyia
$\dfrac n 6$ went to Eurynome
$106$ went to the three Graces
$9 \times 9$ went to the Muses
$7$ remain unpicked.


So we have:

\(\ds n\) \(=\) \(\ds \dfrac n 5 + \dfrac n 8 + \dfrac n 4 + \dfrac n 7 + \dfrac n 6 + 106 + 9 \times 9 + 7\)
\(\ds \leadsto \ \ \) \(\ds 840 n\) \(=\) \(\ds 168 n + 105 n + 210 n + 120 n + 140 n + 840 \times 194\) multiplying through by $840 = \lcm \set {5, 8, 4, 7, 6}$ and simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {840 - 168 - 105 - 210 - 120 - 140} n\) \(=\) \(\ds 840 \times 194\)
\(\ds \leadsto \ \ \) \(\ds 97 n\) \(=\) \(\ds 840 \times 194\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \frac {840 \times 194} {97}\)
\(\ds \) \(=\) \(\ds 1680\)


So there were $1680$ walnuts on the tree, of which:

$336$ went to Parthenopea
$210$ went to Philinna
$420$ went to Aganippe
$240$ went to Orithyia
$280$ went to Eurynome

and as we know:

$106$ went to the three Graces
$81$ went to the Muses

and $7$ remain unpicked.

$\blacksquare$


Source of Name

This entry was named for Metrodorus.


Sources

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