Diophantus died at 84, according to his epitaph. The text never states the age: it splits it into fractions of the life itself, so to recover it you write an equation, multiply it by the least common multiple of the denominators and solve. Here are the setup, a stage-by-stage check and two similar riddles to practice.
What the epitaph says
The epitaph describes the life of Diophantus as a chain of stages measured against one number, the age x at which he died. Childhood took 1/6 of that life, his beard took another 1/12 to grow, and the wedding came after a further 1/7. Five years later a son was born, who lived half as long as his father. The father died four years after the son.
| Stage | Expression | Years with x = 84 | Cumulative age |
|---|---|---|---|
| Childhood | x/6 | 14 | 14 |
| Until his beard grows | x/12 | 7 | 21 |
| Until the wedding | x/7 | 12 | 33 |
| Until the son is born | 5 | 5 | 38 |
| The son's life | x/2 | 42 | 80 |
| Until his own death | 4 | 4 | 84 |
Four stages are fractions of x and two are fixed numbers of years, 5 and 4. The last cumulative age equals the x we were looking for, and that match is the check on the problem.
How to set up and solve the equation
If the stages cover the whole life, their sum is x. The fraction equation looks like this:
Worked example: the age of Diophantus
x/6 + x/12 + x/7 + 5 + x/2 + 4 = x. What is x?
- Find the common denominator. The least common multiple of 6, 12, 7 and 2 is 84.
- Multiply the whole equation by 84. Every fraction becomes a whole number: 14x + 7x + 12x + 42x + 420 + 336 = 84x.
- Add the terms. 75x + 756 = 84x.
- Collect the x terms. 756 = 9x.
- Divide by 9. x = 84.
Result: Diophantus lived 84 years.
Check: 14 + 7 + 12 + 5 + 42 + 4 = 84, as the table above shows.
Step 2 is the delicate one: every term gets multiplied, including the fixed years and the x on the right side. If one is forgotten, the equality no longer holds.
A route without an equation
The four fractions together make 1/6 + 1/12 + 1/7 + 1/2 = 25/28 of the life. What is missing to reach the whole is 1 − 25/28 = 3/28, and that is exactly the 9 fixed years (5 + 4). If 3/28 of the life is 9 years, the whole life is 9 ÷ 3/28 = 84.
This reasoning checks the result by a second route. It also explains why the epitaph works with whole ages: x must be a multiple of 6, 12, 7 and 2 at once, and the smallest common multiple is 84. Any age that is not a multiple of 84, such as 70 or 80, would leave stages with fractions of a year.
What is known about the epitaph
The epitaph is preserved in the Greek Anthology, a collection of epigrams that, according to the MacTutor biography, Metrodorus compiled around the year 500. That same source warns that the details the text gives “may be totally fictitious.” English Wikipedia agrees on the essentials: the epitaph is the only source on the life of Diophantus, and the accuracy of its information cannot be confirmed.
The firmest evidence about his era comes from citations. Diophantus quotes Hypsicles, who wrote after 150 BC, and Theon of Alexandria quotes a definition from Diophantus, so he wrote no later than about 350 AD. Roughly five hundred years lie between those dates. That is why 84 is the solution to a riddle, not a documented age.
Two riddles to practice
They are solved the same way: one unknown, each sentence turned into a fraction of that unknown, and an equation with no denominators. If you want more problems like these, the fraction word problems page collects several types.
Practice
- A teacher spent 1/4 of her life studying, 1/5 teaching in a school and 1/3 training other teachers. She spent her last 13 years retired. How many years did she live?
- A watchmaker’s age is 3/2 of his son’s age, and the watchmaker had his son at age 24. How old is each of them?
Show the answers
- 60 years. The three fractions add to 47/60 of her life, 13/60 is left, and that is the 13 years of retirement, so 13 ÷ 13/60 = 60. Check: 15 + 12 + 20 + 13 = 60.
- The son is 48 and the watchmaker is 72. If y is the son’s age, y + 24 = (3/2) · y, so 24 is half of y. Check: 72 − 48 = 24.
A similar problem, about sharing instead of ages, is the 17 camels puzzle: it also works out when the fractions in the statement do not add up to a whole.
Sources
- Wikipedia, Diophantus: the epitaph as the only source on his life, the sum of its terms giving 84, and the range of possible dates (consulted October 17, 2026).
- J. J. O’Connor and E. F. Robertson, Diophantus of Alexandria, MacTutor History of Mathematics, University of St Andrews: the epigram, the age of 84, and the dating of Diophantus (consulted October 17, 2026).