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Intriguing symmetries are uncovered regarding magic squares of size 3 and 4,and associative magic squares of size 5. In analogy with the travellingsalesman problem, the distributions of the total topological distances of thepaths travelled by passing through all the vertices (matrix elements) only onceand spanning all elements of the matrix are analyzed. Symmetries are found tocharacterise the distributions of the total topological distances in theseinstances. These results open the question of whether these symmetries arefound in higher order magic squares as well.

A Travelling Salesman Paths within nxn (n = 3, 4, 5) Magic Squares