The Afyan Mixed-Grouped Invariant Digit Root (AMGIDR)

This article introduces the Afyan Mixed-Grouped Digit Root Invariant (AMGIDR), a rigorous formulation of an invariance property of digit roots in base 10. Given any positive integer NNN, the paper proves that every regrouping obtained by arbitrarily permuting its digits, cutting them into blocks, interpreting each block as an integer, and summing these integers has the same digit root as NNN, provided each original digit is used exactly once. The result is stated both as a compact “Afyan Principle” and in an explicit mixed-grouping framework, with a detailed proof based on congruences modulo 9. The paper further generalizes the phenomenon to other bases bbb and moduli mmm satisfying b≡1(modm)b \equiv 1 \pmod mb≡1(modm), and discusses potential applications in teaching modular arithmetic, designing recreational problems, and viewing digit roots as simple invariants under structured transformations of numbers.