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 $n$.