Think of a Number/Examples/Rhind Papyrus 28

Example of Think of a Number puzzles

Problem $28$ of the Rhind Papyrus is as follows:

$\dfrac 2 3$ is to be added.

$\dfrac 1 3$ is to be subtracted.

There remains $10$.

Solution

This can more clearly be expressed as:

I think of a number.

I add to it $\dfrac 2 3$ of the number.

I then subtract $\dfrac 1 3$ of the sum.

My answer is $10$.

What number did I think of?

The number was $9$.

Proof

Let $x$ be the number first thought of.

We have:

\(\ds x + \dfrac 2 3 x - \dfrac 1 3 \paren {x + \dfrac 2 3 x}\)

\(=\)

\(\ds 10\)

\(\ds \leadsto \ \ \)

\(\ds x \paren {1 + \dfrac 2 3 - \dfrac 1 3 - \dfrac 2 3 \times \dfrac 1 3}\)

\(=\)

\(\ds 10\)

\(\ds \leadsto \ \ \)

\(\ds x \paren {1 + \dfrac 1 3 - \dfrac 2 9}\)

\(=\)

\(\ds 10\)

\(\ds \leadsto \ \ \)

\(\ds x \times \dfrac {10} 9\)

\(=\)

\(\ds 10\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds 9\)

$\blacksquare$

Sources

• c. 1650 BCE: Ahmes: Rhind Papyrus: Problem $28$
• 1923: T. Eric Peet: The Rhind Mathematical Papyrus: Problem $28$
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Think of a Number: $7$