# Epitaph of Diophantus

## Classic Problem

This tomb holds Diophantus. Ah, how great a marvel!
the tomb tells scientifically the measure of his life.
God granted him to be a boy for the sixth part of his life,
and adding a twelfth part to this, he clothed his cheeks with down;
He lit him the light of wedlock after a seventh part,
and five years after his marriage He granted him a son.
Alas! late-born wretched child;
after attaining the measure of half his father's life, chill Fate took him.
After consoling his grief by this science of numbers for four years he ended his life.

## Solution

Diophantus died at the age of $84$.

Let $x$ be the number of years achieved by Diophantus at his death.

His boyhood took up $\dfrac x 6$ years.

His adolescence took up another $\dfrac x {12}$ years.

After another another $\dfrac x 7$ years he married.

A son was born to him after another $5$ years.

After another $\dfrac x 2$ years, that son died.

(The assumption being made here is the conventional one: that the age of the son at his death is half the age of Diophantus at the death of Diophantus himself, not of his son, which was $4$ years earlier.)

After another $4$ years, Diophantus himself died.

Thus we have:

 $\ds x$ $=$ $\ds \dfrac x 6 + \dfrac x {12} + \dfrac x 7 + 5 + \dfrac x 2 + 4$ $\ds$ $=$ $\ds \dfrac {14 x} {84} + \dfrac {7 x} {84} + \dfrac {12 x} {84} + \dfrac {42 x} {84} + 9$ $\ds$ $=$ $\ds \dfrac {\paren {14 + 7 + 12 + 42} x} {84} + 9$ $\ds$ $=$ $\ds \dfrac {75 x} {84} + 9$ $\ds \leadsto \ \$ $\ds 84 x$ $=$ $\ds 75 x + 9 \times 84$ $\ds \leadsto \ \$ $\ds 9 x$ $=$ $\ds 9 \times 84$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds 84$

Hence Diophantus:

was a boy for $14$ years
was a youth for $7$ years
married after another $12$ years at the age of $33$
had a son born $5$ years later at the age of $38$
who died $42$ years later when Diophantus was $80$
and died $4$ years later at the age of $84$.

$\blacksquare$

## Also known as

This problem is often referred to as Diophantus's riddle.

## Source of Name

This entry was named for Diophantus of Alexandria.

## Historical Note

Whether the Epitaph of Diophantus was actually posed by Diophantus himself is unlikely.

The puzzle seems first to have appeared in the The Greek Anthology Book XIV.

There are a number of translations that can be found. The one given here is that provided by W.R. Paton.
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