Alex’s Puzzling Papyrus

(In the last episode, Dr. Fibonacci Jones discovered a torn scrap of papyrus, covered with hieroglyphic numbers. He promised to teach his daughter, Alexandria, how the ancient Egyptian scribes worked multiplication problems using only the times-two table.)

Alexandria JonesBack at their tent, Dr. Jones handed the papyrus scrap to Alexandria. “What do you see?” he asked.

“Well, there are two columns of numbers,” Alex said. “Let me write them down.” She got a piece of notebook paper and translated the hieroglyphs. Papyrus fragment

Click on the image for a larger view. Translate the numbers for yourself before reading on. If you need help, read Egyptian Math in Hieroglyphs.

What Does It Mean?

“I’ve got the numbers,” Alex said.

\cdot  \; \; 1 \; \; \; 25

\cdot  \; \; 2 \; \; \; 50

\cdot  \; \; 4 \; \; \; 100

\cdot  \; \; 8 \; \; \; 200

\cdot \, \: 13 \; \; 325

[Editor’s note: The dots are there to make the columns line up. Please ignore them, or imagine that they are flecks of dust on the original papyrus calculation.]

She pointed to the papyrus and asked, “But what are these other marks?”

“The scribe marked some of the numbers to aid his calculation,” her father said. “Think of them as checkmarks.”

“Okay,” she said. “Then here is my translation:

\checkmark  \;  1 \; \; \; 25

\cdot  \; \;  \; 2 \; \; \; 50

\checkmark  \;  4 \; \; \; 100

\checkmark  \;  8 \; \; \; 200

\cdot  \;  \; 13 \; \; \, 325

“Hmm,” she continued. “As I go down a column, each number is double the number on the line before. Except for the last line.”

Alex frowned. “Why is the last line different?”

Dr. Jones laughed. “When you work a multiplication problem, what do you put on the last line?”

“The answer?”

“That’s right,” he said.

How Does the Method Work?

Dr. Jones explained, “The scribe who wrote this calculation wanted to multiply:

25 \times 13 = ?

“Now, he could think of that multiplication problem two ways:

How many are 25 thirteens?
— or —
How many are 13 twenty-fives?

“As you know, either question will lead to the same answer. Our scribe decided that 13 twenty-fives would be easier to count.” Dr. Jones pointed to the first line.

\checkmark \; 1 \; \; \; 25

\cdot \; \; \; 2 \; \; \; 50

\checkmark \; 4 \; \; \; 100

\checkmark \; 8 \; \; \; 200

\cdot \; \: 13 \; \; \, 325

“So right here, our scribe started with one 25, then he doubled it (two 25s make 50). He doubled that again (four 25s make 100), and again (eight 25s make 200). He stopped there, because he could see that the next row would have put him over his goal of thirteen 25s.”

“I get it!” said Alex. “Then he marked the rows he needed. One 25, plus four 25s, plus eight more 25s would give him a total of 13 times 25. All he has to do is add up the answer:

Thirteen \; 25s = \left( one \; 25 \right) + \left( four \; 25s \right) + \left( eight \; 25s \right)

13 \times 25 = 25 + 100 + 200 = 325

Alex Tries Another Example

She looked up at her dad. “But does it always work?”

“Yes,” he said. “The scribe’s method will work for any whole numbers. Try it for yourself. Start with an easy problem — how would an Egyptian scribe calculate:

12 \times 5 = ?

Alex thought for a moment. “Let’s see: either 12 fives or 5 twelves… If I count up the 12s, it won’t take as many rows to count five of them.”

She made a chart:

\cdot \; \; 1 \; \; \; 12

\cdot \; \; 2 \; \; \; 24

\cdot \; \; 4 \; \; \; 48

“The next row would be eight 12s, which is more than I want. Now I’ll mark the rows for 1 + 4 = 5, and add up my answer:

\checkmark \; 1 \; \; \; 12

\cdot \; \; \; 2 \; \; \; 24

\checkmark \; 4 \; \; \; 48

\cdot \; \; \; 5

So five 12s make a total of 12 + 48 = 60.”

\checkmark \; 1 \; \; \; 12

\cdot \; \; \; 2 \; \; \; 24

\checkmark \; 4 \; \; \; 48

\cdot \; \; \; 5 \; \; \; 60

Dr. Jones nodded. “Egyptian scribes could multiply almost any two numbers, just by doubling them.”

Alex pouted. “The why did you make me learn my times tables?”

Bigger Numbers Take More Lines

To be sure that Alex understood, Dr. Jones made her work through the steps one more time. He asked her how an ancient Egyptian scribe would calculate this multiplication problem:

69 \times 37 = ?

Step 1: Decide how to count. Shall we figure out what are thirty-seven 69s, or shall we count sixty-nine of the 37s? Alex decided to count up 37 of the 69s.

Step 2: Start with one of whatever you are counting, then double it over and over and over again.

\cdot \; \; 1 \; \; \; 69

\cdot \; \; 2 \; \; \; 138

\cdot \; \; 4 \; \; \; 276

\cdot \; \; 8 \; \; \; 552

\cdot \, \; 16 \; \; 1104

\cdot \, \; 32 \; \; 2208

The next row would count 64 of the sixty-nines, but Alex only wanted 37 of them. So she stopped doubling and went on to…

Step 3: Mark the numbers in the first column that add up to however many you need to count. Alex needed to count 37 of the sixty-nines, and:

32 + 4 + 1 = 37

So she marked her chart like this:

\checkmark \; 1 \; \; \; 69

\cdot \; \; \; 2 \; \; \; 138

\checkmark \; 4 \; \; \; 276

\cdot \; \; \; 8 \; \; \; 552

\cdot \; \; \, 16 \; \; 1104

\checkmark \: 32 \; \; 2208

\cdot \; \; \, 37

Step 4: Add up the marked numbers in the second column. That is, take the second-column number from each marked row and add them together to get your answer. Alex added up:

69 + 276 + 2208 = 2553

Her final chart looked like this:

\checkmark \; 1 \; \; \; 69

\cdot \; \; \; 2 \; \; \; 138

\checkmark \; 4 \; \; \; 276

\cdot \; \; \; 8 \; \; \; 552

\cdot \; \; \, 16 \; \; 1104

\checkmark \: 32 \; \; 2208

\cdot \; \; \, 37 \; \; 2553

Now You Try It

You will never remember a math technique just by reading about it. In order to learn, you have to do it yourself. Math is NOT a spectator sport!

So make up some multiplication problems of your own, and try to solve them the Egyptian way. Use whole numbers to keep it simple — no fractions or decimals. After you get your answers, check them by multiplying the standard way that you learned in school.

You decide: Which method is easier? Why?

The Egyptian scribe’s method of multiplication will work for fractions as well, but I think this is enough for today, don’t you? We will talk about Egyptian fractions another time….

Edited to add: More math translation and multiplication practice awaits you in next week’s installment — Egyptian math puzzles.

To Be Continued…

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


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4 thoughts on “Alex’s Puzzling Papyrus

  1. So they’ve just doubling…
    Looking on the left column I see a sequence of powers of two. And these checkmarks which selects the used ones is just like a *binary code*. Am I right?
    So what does this binary code mean here? I see it encodes the multiplier. But is it used in some other way too?

  2. The binary code here is counting how many of the multiplicand you have. When the count makes the multiplier, then the sum of those lines in the second column will make the product of the two numbers you want.

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