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In this study, we embark on a mathematical exploration of 4x4 magic squares, delving into the intriguing symmetries hidden within these enigmatic structures. Our focus lies on a traveler's quest to visit cities positioned at the center of each cell in a 4x4 square grid, following the sequence dictated by a magic square arrangement. However, the distances between consecutive cities lack the magical properties associated with magic squares. Through meticulous calculations, we analyze the distances between cities, unraveling fascinating symmetrical patterns along the traveler's journey. Our findings shed light on the remarkable symmetries intrinsic to magic squares, transcending disciplines such as mathematics, physics, and computer science. This exploration opens avenues for further investigations into the captivating realm of symmetry, inspiring new discoveries and deepening our understanding of these intriguing mathematical constructs., Comment: 8 pages, 4 figures