magicnumerot

Magicnumerot is a term used in some puzzle communities and educational contexts to denote integers that satisfy a chosen invariant rule under a specified transformation. The expression blends the idea of “magic” with numerot, a form related to the word for numbers in several languages, and is not a standardized mathematical term. In practice, the definition of a magicnumerot depends on the rule and the transformation chosen for a given problem or discussion.

Definition and scope: A magicnumerot is defined by a rule R that depends on a particular representation or encoding of numbers. Given a transformation T that acts on that representation (for example, a digit operation, a base-b expansion, or a modular reduction), a number n is called a magicnumerot if T(n) equals n, or if repeated application of T yields a fixed point or a pre-specified short cycle. Because different contexts adopt different rules, the set of magicnumerot can vary widely between problems.

Construction and examples: To construct magicnumerot, one selects a base and a transformation and then solves for n such that T(n) = n (or T^k(n) = n for some chosen k). Some families are straightforward to enumerate, while others require search or algorithmic techniques. In many presentations, examples are given with small numbers or framed as challenges to find all magicnumerot below a bound.

Relation and context: Magicnumerot relate conceptually to fixed points and invariants in number theory, and they intersect with ideas such as automorphic numbers, Kaprekar-type properties, and digital-root invariants. Because there is no universal definition, readers should refer to the specific rule and transformation used in a given discussion.