# Rhind Papyrus 100 Loaves 5 Men Puzzle - Solution

## The Puzzle:

About 2000 years BC there lived Ahmes, a royal secretary and mathematician of the Pharaoh Amenemhat III. One of his papyruses was found in 1853 by an Englishman called Rhind near the temple of Ramses II in Thebes. It has many mathematical puzzles and here is one:

100 loaves of bread must be divided among five workers.

Each worker in line must get more than the previous: the same amount more in each case (an arithmetical progression).

And the first two workers shall get seven times less than the three others.

How many loaves (including fractions of a loaf!) does each worker get?

100 loaves of bread must be divided among five workers.

Each worker in line must get more than the previous: the same amount more in each case (an arithmetical progression).

And the first two workers shall get seven times less than the three others.

How many loaves (including fractions of a loaf!) does each worker get?

## Our Solution:

Let us say the middle worker (worker 3) gets "w" loaves.

And that "d" is the common difference between workers.

So the workers get:

w-2d

w-d

w

w+d

w+2d

The middle worker gets a perfect average, so 100/5 = 20 loaves

The first two workers get seven times less than the three others:

7*[(20-2d) + (20-d)] = 20 + (20+d) + (20+2d)

From this: d = 220/24, or 55/6

And this is the solution:

1st worker = 10/6 loaves

2nd worker = 65/6 loaves

3rd worker = 120/6 (20) loaves

4th worker = 175/6 loaves

5th worker = 230/6 loaves

And that "d" is the common difference between workers.

So the workers get:

w-2d

w-d

w

w+d

w+2d

The middle worker gets a perfect average, so 100/5 = 20 loaves

The first two workers get seven times less than the three others:

7*[(20-2d) + (20-d)] = 20 + (20+d) + (20+2d)

From this: d = 220/24, or 55/6

And this is the solution:

1st worker = 10/6 loaves

2nd worker = 65/6 loaves

3rd worker = 120/6 (20) loaves

4th worker = 175/6 loaves

5th worker = 230/6 loaves