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Math Trek

Magic Square Physics

3:57pm, June 30, 2006
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The study of magic squares has a long, long history. In ancient Babylonian times, these array of numbers were held to have magical powers. Over the years, they have also served as protective charms and religious symbols.

At the recreational level, they're fun for all ages, says Peter Loly of the University of Manitoba.

Loly himself has investigated the "physical" properties of magic squares—treating the numbers of each such square as physical quantities.

A magic square is a square matrix drawn as a grid filled with numbers. It consists of a set of integers arranged in the form of square so that the numbers in each row, column, and diagonal all add up to the same total. If the integers are consecutive numbers from 1 to n2, the square is said to be of nth order. The magic sum itself is given by n(n2 + 1)/2.

Suppose, for example, 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 mr2 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, Iz, 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.

In general, you can show that, for order n, Iz = [n2 + (n4 – 1)]/12.

So, for n = 3, Iz = 60; for n = 4, Iz = 340.

"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.

The same formula applies to semi-magic squares, which don't meet the standard requirement that diagonals also sum to the magic number.

Such an analysis can be extended to magic cubes. A magic cube consists of n3 numbers, arranged so that each row, column, and main diagonal give the same sum. In the case, the magic constant is n(n3 + 1)/2.

Here's one example.


The three layers (above) of a 3-by-3 magic cube.

Loly and Rogers show that the moment of inertia of a magic cube is n3(n3 + 1)(n2 – 1)/12. In effect, they demonstrate that magic cubes have the same inertial form as a spherical top.

It's also possible to consider the numbers of magic squares to be electric charges and to extend such analyses to higher dimensions. Loly has even calculated the so-called eigenvalues of magic squares, which are related to their "fundamental frequencies" if you were to set these squares ringing like a bell.

"When treated as mass distributions, magic squares give clear and accessible examples of the properties of the moment of inertia," Loly says. "When treated as matrices, magic squares also serve as exceptional examples of some advanced linear algebra theorems."

Check out Ivars Peterson's MathTrek blog at

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