Myths
The Farmer’s Will
Story Summary
Before dying, a farmer leaves his three sons a will in which he explains how he wishes his 17 horses to be divided among them after his death. His unequal division leaves the heirs puzzled and forces them to seek help from an old farmer. Cleverly, this farmer adds his own horse to the count and manages to satisfy each son beyond their expectations, all while keeping his own horse!
Source: tale adapted from the story of the 17 camels, attributed to the Arab mathematician Alî ibn Abî Tâlib, a 7th-century figure and skilled master of logical reasoning.
About the Story
This tale poses a mathematical problem: how to divide 17 by two, by three, and by nine to obtain whole numbers?
The family setting of the problem makes the riddle entertaining and its solution important to find. The old farmer’s trick allows the sons to receive more than what was expected: it’s a great way to encourage students to take an interest in fractions. The issue of indivisibles suddenly becomes a matter of personal gain!
👉 Discover the tale
Maths
Proportions, Disputes…
and a Horse that Didn’t Count
👦🏻 Target Age: 8–10 years (Grades 3–5)
⏰ Estimated Duration:
🔢 Mathematical Concepts Involved
- Fractions
- Fractions of a quantity
- Euclidean division with remainder
- Solving a problem involving sharing
🎯 Learning Objectives
Develop the following mathematical skills:
- Calculating
- Reasoning
- Modeling
Develop the following life skills (psychosocial competencies):
- Critical thinking
- Problem solving
- Creative thinking
- Strategy
💬 Teaching Note
This story is very effective in creating mathematical questioning: a horse is added only for the time of the division, its presence solves all the calculations, and yet that added horse can be taken back! This questioning is an ideal starting point for learning.
🟢 Activity 1
- Carry out the Euclidean divisions
$17 \div 2$ $17 \div 3$ $17 \div 9$
Explain why the division is problematic.
💬 Teaching Note
17 divided by 2, the quotient is 8, remainder 1, so $17 = 2 \times 8 + 1$
$\begin{array}{r|r}17 & 2 \\-16 & 8 \\\hline1 &\end{array}$
17 divided by 3, the quotient is 5, remainder 2. $17 = 3 \times 5 + 2$
$\begin{array}{r|r}17 & 5 \\-15 & 3 \\\hline2 &\end{array}$
17 divided by 9, the quotient is 1, remainder 8. $17 = 9 \times 1 + 8$
$\begin{array}{r|r}17 & 9 \\-9 & 1 \\\hline8 &\end{array}$
Each of the “shares” suggested by the will is actually impossible to achieve because 17 is divisible neither by 2, nor by 3, nor by 9. With 17 horses, one might propose giving 8 horses to the first son, 5 to the second, and 1 to the last; there would remain $17 - (8 + 5 + 1) = 3$, clearly nothing works. One can also attempt decimal division and see that the results are not whole numbers. How can we resolve this contradiction?
🟢 Activity 2
- Calculate
$18 \div 2$ $18 \div 3$ $18 \div 9$
- Add the three results you obtained
- What do you notice?
💬 Teaching Note
If we add all the shares as fractions, we don’t get 1. Depending on students’ progress, we may reason with whole numbers, summing to 17 and not 18, or with fractions that don’t add up to a whole:
$9 + 6 + 2 = 17$
$\frac{1}{2} + \frac{1}{3} + \frac{1}{9} = \frac{9}{18} + \frac{6}{18} + \frac{2}{18} = \frac{17}{18}$
This means that in his will, the farmer is actually not dividing all of his possessions!