# Basic Algebra Puzzle

1. Nov 10, 2006

### AKG

"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.

2. Nov 10, 2006

### 0rthodontist

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: Nov 10, 2006