You can’t give half, a third, and a ninth of 17 camels without cutting one up. The traveler who lends his own camel solves it: with 18, the shares come out to 9, 6, and 2, and the borrowed camel goes back to its owner. There is no trick in the arithmetic. The will only hands out 17/18 of the herd, and the 1/18 nobody receives is exactly the spare camel.
The puzzle is a traditional sharing story with many versions. Below you will see which fractions make it work and why each brother comes out ahead.
Worked example: sharing the 17 camels
A father leaves 17 camels: half to the eldest son, a third to the middle son, and a ninth to the youngest. A traveler adds his own camel. How many does each son get?
- Add the borrowed camel. There are 17 + 1 = 18 camels.
- Find the eldest’s share. Half of 18 is 9.
- Find the middle son’s share. A third of 18 is 6.
- Find the youngest’s share. A ninth of 18 is 18 × 1/9 = 2.
- Add up what was handed out. 9 + 6 + 2 = 17, so one camel is left.
Result: 9, 6, and 2 camels, and the traveler takes his own back.
Check: the whole-number shares add up to 17, exactly the real herd. Nobody had to cut up an animal.
Why one camel is left over
The reason is in the sum of the fractions in the will. First add 1/2 + 1/3 = 5/6. Rewrite 5/6 in eighteenths, which gives 15/18, and add 1/9 = 2/18: the total is 17/18. If you want to check a step, the adding fractions calculator shows each common denominator.
The three fractions add up to less than 1. The father left 1/18 of the herd unassigned. With 18 camels, 1/18 is one whole camel, the one that goes back. With 17 camels, that part is 17/18 of an animal, something nobody can receive as it stands.
Each brother gains in proportion
Applied to 17 camels, the will would give shares with fractions. With the loan, each brother gets more than that.
| Son | Share written | Receives | Gains |
|---|---|---|---|
| Eldest | 17/2 = 8 1/2 | 9 | 1/2 |
| Middle | 17/3 = 5 2/3 | 6 | 1/3 |
| Youngest | 17/9 = 1 8/9 | 2 | 1/9 |
Each gain is 1/17 of the written share: the eldest’s gain of one half is a seventeenth of 8.5. That is why the result respects the proportions: 9 : 6 : 2 is the same as half : third : ninth, with everything multiplied by 18. Nobody loses out, and each son receives 18/17 of what he was owed.
The same split without a borrowed camel
Split the 17 camels in the ratio 9 : 6 : 2, which adds up to 17. The eldest gets 17 × 9/17 = 9, the middle son 6, and the youngest 2. The traveler’s camel is a way to reach that ratio without doing the math.
For other totals, the proportional sharing calculator does this with weights written as fractions. The guide to proportional sharing with fractions explains how to set it up.
When the puzzle works
The trick needs the fractions to add up to n/(n + 1), with n camels and one borrowed. In addition, each denominator must divide n + 1, so that the shares come out whole.
| Camels | Fractions | With one borrowed | Shares |
|---|---|---|---|
| 7 | half, 1/4, 1/8 | 8 | 4, 2, 1 |
| 11 | half, 1/4, 1/6 | 12 | 6, 3, 2 |
| 19 | half, 1/4, 1/5 | 20 | 10, 5, 4 |
| 23 | half, 1/3, 1/8 | 24 | 12, 8, 3 |
Check the row for 11: 6/12 + 3/12 + 2/12 = 11/12, and 12 is a multiple of 2, 4, and 6. With 35 camels and the fractions of the original problem, the split of 36 is 18 + 12 + 4 = 34 and two are left over, so returning one no longer settles the account.
If the fractions added up to exactly 1, as in 1/2 + 1/3 + 1/6, nothing would be left over and there would be no loan to return.
Where it comes from
In the Shia tradition the story is attributed to Imam Ali, and the historian Pierre Ageron studied how that attribution spread and how the tale reached Europe. His article covers the history of the story. The arithmetic of the split is the one above, whoever tells it.
Sources
- Pierre Ageron, “Le partage des dix-sept chameaux et autres exploits arithmétiques attribués à l’imâm ʿAlî,” Revue d’histoire des mathématiques 19(1), 2013, pp. 1-41 (in French). The Société mathématique de France page carries the abstract: doi.org/10.24033/rhm.169 (accessed October 16, 2026).
- Spanish Wikipedia, article on fractions, section “Suma y resta de fracciones,” on the common denominator and the least common multiple (in Spanish; accessed October 16, 2026).