Complex Magic Squares and the Squared Gaussian Integer Lattice

SummaryMagic squares are grids of integer numbers where each row, column, and diagonal sums to the same constant. Magic squares have fascinated mathematicians for over two millennia but despite their long history, aspects of real and complex magic squares still remain undiscovered. We present new findings on the nonexistence of third order squared magic squares consisting of squared Gaussian integers using a geometric approach between magic squares and parallelograms in the complex plane. Nonexistence proofs are presented in the squared Gaussian integer lattice for axis-aligned origin-centered rectangles and rhombi which translate to magic squares. The geometric relation also extends the presented proofs to other polygons in the complex plane. For the first time, multiplicative complex squared magic squares are identified. Infinite solution spaces are presented for three unique multiplicative squares. These findings aim to develop the understanding of complex squared magic squares and the squared Gaussian integer lattice by introducing new constraints and possibilities.