Definition:Think of a Number
Definition
A think of a number puzzle is usually in the form of a game between two players.
- 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:
- $(2): \quad$ 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: $1$
- 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.