# Solve Abu Kamil's Algebra Puzzle from 9th/10th Century

• AKG
In summary, the conversation discusses a problem by Islamic mathematician Abu Kamil, which was found in a review of the book "Unknown Quantity" about the history of algebra. The problem involves dividing the number 10 into three parts and using multiplication to solve for the values of each part. The solution involves using variables and equations to represent the different parts and solve for their values.
AKG
Homework Helper
"One says that ten is divided into three parts, and if the small one is multiplied by itself and added to the middle one multiplied by itself, it equals the large one multiplied by itself, and when the small is multiplied by the large, it equals the middle multiplied by itself."

This is a problem of 9th/10th century Islamic mathematician Abu Kamil. I found it in a an issue of Science in a review of "Unknown Quantity", a book about the history of algebra. It kept me entertained while I was waiting for a professor, see if you can figure it out. On a side note, the author's response to the review of his book, along with a reproduction of the review itself, are here.

a <= b <= c
a + b + c = 10
aa + bb = cc
ac = bb
aa + ac = cc
a/c + 1 = c/a
let x = a/c
x + 1 = 1/x
xx + x - 1 = 0
x = (-1 + sqrt(1 + 4))/2 = a/c
xcc = bb
b = sqrt(x)c
cx + sqrt(x)c + c = 10
c = 10/(x + sqrt(x) + 1)
a = cx
b = sqrt(x)c

Last edited:

After examining the puzzle, it seems that we are being asked to find three numbers that satisfy the given conditions. Let's start by assigning variables to the three parts mentioned in the puzzle. We can call the small part "x," the middle part "y," and the large part "z."

Based on the given information, we can create the following equations:

x + y + z = 10 (since the parts add up to 10)
x^2 + y^2 = z^2 (since the sum of the squares of the small and middle parts equals the square of the large part)
x * z = y^2 (since the product of the small and large parts equals the square of the middle part)

Now, we have a system of three equations with three variables. We can use algebraic techniques to solve for the values of x, y, and z.

First, let's rearrange the second equation to isolate z^2:

z^2 = x^2 + y^2

Then, substitute this expression for z^2 into the third equation:

x * z = y^2
x * (x^2 + y^2) = y^2
x^3 + xy^2 = y^2

Next, rearrange the first equation to isolate z:

z = 10 - x - y

Substitute this expression for z into the second equation:

x^2 + y^2 = (10 - x - y)^2
x^2 + y^2 = 100 - 20x - 20y + x^2 + 2xy + y^2
0 = 100 - 20x - 20y + 2xy

Finally, we can solve for y in terms of x by rearranging this equation:

20y = 100 - 20x + 2xy
y = (100 - 20x + 2xy)/20
y = 5 - x + (xy)/10

Now, we can substitute this expression for y into the first equation:

x + (5 - x + (xy)/10) + (10 - x - (xy)/10) = 10
5 - x + (xy)/10 = 0

Solving for x, we get x = 5. Then, substituting this value into the expression for y, we get y = 5. Finally,

## What is Abu Kamil's Algebra Puzzle from 9th/10th Century?

Abu Kamil's Algebra Puzzle is a mathematical problem created by the Arab mathematician Abu Kamil Shuja ibn Aslam in the 9th/10th century. It involves solving a system of equations with multiple unknown variables.

## Why is Abu Kamil's Algebra Puzzle important?

Abu Kamil's Algebra Puzzle is important because it is one of the earliest known examples of a mathematical problem that involves solving a system of equations. It also demonstrates the advanced level of mathematical knowledge and skills present in the Arab world during the 9th/10th century.

## How do you solve Abu Kamil's Algebra Puzzle?

To solve Abu Kamil's Algebra Puzzle, you need to identify the unknown variables and set up a system of equations using the given information. Then, you can use algebraic techniques such as substitution or elimination to solve for the unknown variables and find the solution to the puzzle.

## What can we learn from solving Abu Kamil's Algebra Puzzle?

Solving Abu Kamil's Algebra Puzzle can help us understand the development of algebra and mathematical problem-solving strategies in the 9th/10th century. It also showcases the creativity and ingenuity of early mathematicians in solving complex problems.

## Are there any modern applications of Abu Kamil's Algebra Puzzle?

While Abu Kamil's Algebra Puzzle may not have direct applications in modern mathematics, it serves as a foundation for more advanced algebraic concepts and problem-solving techniques. The principles used to solve this puzzle can be applied to various real-world problems in fields such as engineering, physics, and economics.

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