This study introduces a novel class of elastic metamaterials inspired by magic squares, mathematical constructs where the sum of each row, column, and diagonal equals a constant "magic number." Leveraging these principles, we create "magic metamaterials" that allow for tunable elastic wave bandgaps without altering the overall mass, stiffness, or moment of inertia. We demonstrate this through three configurations of spring-mass supercells following magic square mass distributions. In Configuration 1, interactions are limited to nearest neighbors with horizontal and vertical springs; Configuration 2 incorporates additional internal interactions via diagonal springs connecting central and corner masses within each supercell; and Configuration 3 introduces further complexity with diagonal connections between supercells and next-nearest neighbor interactions. Dispersion analyses of each configuration reveal that wave attenuation and bandgap properties are tunable while maintaining global physical properties. To validate this approach, we developed a physical model using spring-mass systems arranged in a lattice embedded with magic square masses and used finite element analysis (FEA) in Abaqus to obtain eigenfrequencies and out-of-plane polarized wave attenuation bandgaps. Furthermore, plate structures, including those embedded with magic square masses and those with mounted resonators, are analyzed, with results closely aligning with idealized models, confirming that the dynamic behavior of these structures can be adjusted while preserving mass and stiffness. This study demonstrates the potential of magic square-based designs for robust, tunable wave manipulation in elastic metamaterials.