# Diophantus II

Achievement Objectives
NA6-5: Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns.
Student Activity

Mathematical puzzles have fascinated people throughout the ages.
These were often expressed in verse or as riddles like this:

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 of his father’s full life, chill Fate took him.
After consoling his grief by this science of numbers for four years he ended his life.

How long did Diophantus live?

[Diophantus, a Greek mathematician]

Specific Learning Outcomes

solve a problem in a number of ways, including using algebraic expressions

Description of Mathematics

Students may approach this problem using guess and check or by being systematic in some way. However, students should be supported to understand, and to use, an algebraic method of solution, recognising its efficiency.

The problem was apparently engraved on a tombstone in the time of the Greek mathematician Diophantus who lived in Alexandria somewhere between 150 BC and 364 AD. Diophantus wrote a thirteen-volume set of books called Arithmetica of which only six  have survived. He was interested in problems that had whole number solutions. Accordingly, equations of this type are called Diophantine equations.

Other related problems include:  A Lady’s Age and Diophantus I Algebra Level 6.

Required Resource Materials
Activity

### The Problem

Mathematical puzzles have fascinated people throughout the ages. These were often expressed in verse or as riddles like this:

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 of his father’s full life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life.

How long did Diophantus live?

Reported to have been inscribed on the grave of Diophantus, a Greek mathematician from Alexandria (100BC approx)]

### Teaching sequence

1. Have students consider the historical nature of the puzzle with questions such as:
Who is the most famous person you know who was born over 100 years ago?… over 1000 years ago? … over 2000 years ago?
Who was the dominant power in Europe 2000 years ago?
Where was the European centre of learning then?
2. Read Diophantus’ poem and, with the students, tease out it's meaning.
3. Have students work on the problem, recording their working as they do so. Monitor their progress.
4. When a group finds a solution using an algebraic approach, encourage them to work on the Extension problem.
5. Allow time for several groups to share their solutions and have the class discuss these together.
6. Make time for students to write up two ways of solving the problem.

#### Extension to the problem

Write a problem about your own age, or someone else’s age, for members of the class to solve.

Using online sources, find out as much as you can about Diophantus or other ancient Greek mathematicians.

### Solution

Your students may do this a number of ways (see A Lady’s Age and Diophantus I). The algebraic method is given here.

Suppose that Diophantus lived to be d years of age. He was a boy for d/6 years; had to shave after d/12 more years; was married after a further d/7 years; had a son 5 years later; his son died d/2 years later; then Diophantus died 4 years later. So

d = d/6 + d/12 + d/7 + 5 + d/2 + 4 = 75d/84 + 9.

Hence 9d/84 = 9 or d/84 = 1. So d = 84.

Diophantus lived to the age of 84 years.

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