This paper introduces the Mamzeris Method, a novel and systematic approach for constructing associative magic squares of any odd order. Developed by Smaragdos (Marios) Mamzeris in 1988, the method utilises sequential transformations of an N×NN×N grid to achieve the magic constant across rows, columns, and diagonals. The paper presents the step-by-step procedure, explores variations and permutations, and demonstrates its effectiveness with examples. The Mamzeris Method advances recreational mathematics by offering a universal solution for odd-ordered magic squares and highlights the potential applications in mathematical theory, cryptography, and algorithm design.