# 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.

## Sources

- 1918: W.R. Paton:
*The Greek Anthology Book XIV*... (previous) ... (next): Metrodorus' Arithmetical Epigrams: $126$ - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.4$ The Diophantine Equation $a x + b y = c$ - 1980: Angela Dunn:
*Mathematical Bafflers*(revised ed.): $1$. Say it with Letters: Algebraic Amusements: The Ages of Man - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $84$ - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): Metrodorus and the*Greek Anthology*: $35$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $84$

- Weisstein, Eric W. "Diophantus's Riddle." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantussRiddle.html