# Think of a Number/Examples/Rhind Papyrus 33

## Example of Think of a Number puzzles

Problem $30$ of the Rhind Papyrus is as follows:

A number, plus its two-thirds, and plus its half, plus its seventh, makes $37$. What is the number?

It is to be expressed in Egyptian form.

## Solution

This can more clearly be expressed as:

I think of a number.
If you add $\dfrac 2 3$ of that number plus $\dfrac 1 2$ of that number and $\dfrac 1 7$ of that number, you get $37$.
What number did I think of?

The number was $16 + \dfrac 1 {56} + \dfrac 1 {679} + \dfrac 1 {776}$.

## Proof

Let $x$ be the number first thought of.

We have:

 $\ds x + \dfrac 2 3 x + \dfrac 1 2 x + \dfrac 1 7 x$ $=$ $\ds 37$ $\ds \leadsto \ \$ $\ds x \paren {1 + \dfrac 2 3 x + \dfrac 1 2 x + \dfrac 1 7}$ $=$ $\ds 37$ $\ds \leadsto \ \$ $\ds x \paren {\dfrac {42 + 28 + 21 + 6} {42} }$ $=$ $\ds 37$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds 37 \times \dfrac {42} {97}$ $\ds$ $=$ $\ds \dfrac {1554} {97}$ $\ds$ $=$ $\ds 16 + \dfrac 2 {97}$ $\ds$ $=$ $\ds 16 + \dfrac 1 {56} + \dfrac 1 {679} + \dfrac 1 {776}$ from a table of Egyptian fractions

$\blacksquare$