# Egyptian Geometry and Other Challenges

Would you like to study “the knowledge of all existing things and all obscure secrets”? That is how Scribe Ahmose (also translated Ahmes) described his mathematical papyrus. Ahmose’s masterpiece is now called the Rhind Papyrus, after Alexander Henry Rhind, a Scotsman who was one of the first archaeologists to make meticulous records of his excavations (rather than simply hunting for treasures). Rhind purchased the papyrus from an antiquities dealer in Luxor, Egypt, in 1858.

Ahmose’s writing included a huge table of fractions as well as story problems, geometry, algebra, and accounting. Can you solve any of Scribe Ahmose’s problems?

## Ancient Egyptian Geometry

(1) Ahmose calculates the area of a circle as equal to a square with sides that are 8/9 of the circle’s diameter. This is the same as the standard formula for the area of a circle, $A = \pi {r}^{2}$, if we use an approximation for pi. What is Ahmose’s value for pi?

(2) If you keep two sides of a triangle the same but let the other side change, you could make an infinite number of triangles. Of all those possible triangles, which one has the largest area? [Hint: Feel free to experiment with specific triangles, if you like, but it is possible to solve this problem without using any numbers at all.]

## From the Moscow Papyrus

(3) The area of a rectangle is 12, and the width is 3/4 of the length. How long are the sides of the rectangle?

(4) One leg of a right triangle is 2 1/2 times the other, and the area is 20. How long are the triangle’s sides?

## As I Was Going to St. Ives…

(5) Problem #79 of the Rhind papyrus is unusual, and translators are not sure how to interpret it. One translation goes as follows:

A rich man’s estate contained 7 houses.
Each cat ate 7 mice.
Each mouse devoured 7 heads of wheat.
And each head of wheat could yield 7 hekat measures of grain.
Houses, cats, mice, heads of wheat, hekat measures of grain—how many of these were in all the estate?

## Can You Explain This?

Here is a math “magic trick” from the Rhind papyrus. (Warning: The Egyptian scribes loved working with fractions.) Your job is to explain why it works. How could Scribe Ahmose know that he would always be able to tell what number his friend had in mind?

• Tell your friend to think of a secret number.
[To avoid fractions, pick a multiple of 9.]
• Then have him add 2/3 more to his number.
[So if he started with 9, he would add 2/3 of 9: 9 + 6 = 15.]
• Finally, tell him to take away 1/3 of this total, and say the answer.
[1/3 of 15 is 5, and 15 – 5 = 10. Your friend would say, “Ten.”]
• Now you must subtract 1/10 of that number to find the secret.
[1/10 of 10 is 1, so the secret number is 10 – 1 = 9.]

Like the Egyptian scribes, mathematicians through the ages have built up their knowledge by making up puzzles for each other and then solving them. Can you make up a math problem similar to those in Scribe Ahmose’s papyrus? I would love to include your puzzle in a future Alexandria Jones story! Look here for an email form to send in the problem(s) you have made.

[Images copyrighted by Historylink101.com and found at the Egyptian Picture Gallery. Click on any photo for a larger view.]

The answers to these puzzles (and more) are now posted here.

## To Be Continued…

Read all the posts from the September/October 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.

## 5 thoughts on “Egyptian Geometry and Other Challenges”

1. ∏ ≈ 3.1604938 . . .

uh oh . . . do you want to see our work?

2. No need to show me your work. Looks like you got it just fine! I will be posting a list of all the answers for this series on Egyptian math soon—I think I have that scheduled for next week, if school doesn’t get in the way.

Edited to add: Meeyauw did show her work in this post, and she got 72 comments on it. Wow!

3. This is very cool — thank you for this post. And who doesn’t want “knowledge of all existing things?” You can’t fault Ahmose for having unambitious goals. 😉

4. This is so great! We are doing a block on Egypt in October, and I wanted to use Egyptian multiplication. These little puzzles will be great in-between brain teasers. Thanks so much!

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