Definition:Think of a Number

Definition

A think of a number puzzle is usually in the form of a game between two players.

Player A asks player B to:

Think of a number

perhaps with constraints.

Let this number be referred to as $n$.

Player A asks player B to perform certain arithmetical manipulations on $n$.

As a result, player B is left with another number, which we will refer to as $m$.

The game now goes one of $2$ ways:

$(1): \quad$ Player A announces:

The number you have been left with is $m$.

$(2): \quad$ Player A asks what $m$ is, and on learning what it is, instantaneously replies:

The number you first thought of was $n$.

Also known as

Henry Ernest Dudeney refers to such a puzzle as a boomerang, probably because of the way it returns to the asker.

It is believed that this nomenclature is idiosyncratic.

Examples

Rhind Papyrus Problem $28$

$\dfrac 2 3$ is to be added.

$\dfrac 1 3$ is to be subtracted.

There remains $10$.

Rhind Papyrus Problem $30$

If the scribe says to thee:

$10$ has become $\dfrac 2 3 + \dfrac 1 {10}$ of what?

Rhind Papyrus Problem $33$

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.

Bachet: $1$

A person chooses secretly a number, and trebles it, telling you whether the product is odd or even.

If it is even, he takes half of it,

or if it is odd, he adds one and then takes one half.

Next he multiplies the result by $3$,

and tells you how many times $9$ will divide into the answer, ignoring the remainder.

The number he chose is -- what?

Bachet: $2$

The subject chooses a number less than $60$

and tells you the remainders when it is divided by $3$, $4$ and $5$, separately, not successively.

The original number is -- what?

Bachet: $3$

The first person takes a number of counters greater than $5$.

The second person takes $3$ times as many.

The first person gives $5$ counters to the second.

The second person then gives the first $3$ times as many as the first person holds in his hand.

How many counters has the second person got in his hand?

Also see

• Results about think of a number puzzles can be found here.

Historical Note

The think of a number puzzle goes way back in time.

Henry Ernest Dudeney discusses it in his posthumous ($1932$) collection Puzzles and Curious Problems as follows, presented on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a historical note:

In the words of Henry Ernest Dudeney:

One of the most ancient forms of arithmetical puzzle is that which I call the "Boomerang."

Everybody has been asked at some time or another to "Think of a number,"

and after going through some process of private calculation, to state the result,

when the questioner promptly tells you the number you thought of.

There are hundreds of varieties of the puzzle.

The oldest recorded example appears to be that given in the Arithmetica by Nicomachus, who died about the year $120$.

He explains that:

He tells you to think of any whole number between $1$ and $100$, and then divide it successively by $3$, $5$ and $7$, telling him the remainder in each case.

On receiving this information he promptly discloses the number you thought of.

Note, however, that since Dudeney wrote the above, the Rhind Papyrus from was found to contain a number of examples of this puzzle.

This pushes the earliest date back to $\text {c. 1650}$ $\text {BCE}$, considerably earlier than Nicomachus.