| "This is the only property of magic squares, aside from the line sums, which is solely dependent on the order of the square, n," Loly and Adam Rogers note in a paper published in 2004 in the Canadian Undergraduate Physics Journal.
Loly investigated the "physical" properties of magic squares, treating the numbers of each such square as physical quantities. If the integers are consecutive numbers from 1 to n^2, the square is said to be of n-th order. The magic sum itself is given by n(n^2 + 1)/2.
Suppose you interpret the numbers as masses. You can then determine a magic square's moment of inertia about a given axis of rotation. For any specific case, you obtain the moment of inertia, In, of a magic square of order n about an axis at right angles to its center by summing mr^2 for each cell, where m is the number centered in a cell and r is the distance of the center of that cell from the center of the square measured in units of the nearest neighbor distance.
You find that the moment of inertia, I_z, about the square's center (an axis at right angles to the square) is twice the moment of inertia about an axis of rotation along the center row or column.
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