Details

is odd

Define

,

.

Then the magic square is

.

is doubly even (i.e. is divisible by 4)

Define

1. ,

2. ,

3. ; here, "" is the equal operation yielding True or False component-wise, so that is a matrix of Boolean values.

Define to be a matrix of the integers from 1 to , stored sequentially in rows.

The elements in where is False are left alone, and the elements in where is True are reversed; that is, the value at a True position is replaced with .

For example, let

1.

2.

3.

The elements in where is False are left alone.

The elements in where is True are reversed; that is, at such a position with value , the new value is .

So this is final result.

is singly even (i.e. is divisible by 2 but not by 4)

If is singly even, then is odd, and a magic square of order can be constructed from four copies of the magic square of order .

Construct a matrix with four blocks:

,

where is a magic square of order , , , and .

Since is odd, for some . Suppose

.

1. Swap the values of at , , and with the corresponding values of .

Suppose

.

2. Swap the values of at with the corresponding values of .

For instance, let , so .

Then the block matrix (with the first swap colored) is

.

1. Swap the values of at and with the corresponding values of . (Now the second swap is colored.)

2. Swap the values of at with the corresponding values of .

References

[1] C. Moler, Experiments with MATLAB, MathWorks, Inc., 2011 pp. 129–132. www.mathworks.com/moler/exm.

[2] kguler. "Answer to 'Better Method to Swap the Values of Two 2-D Arrays'." Mathematica Stack Exchange. (Apr 23, 2015) mathematica.stackexchange.com/questions/80674/better-method-to-swap-the-values-of-two-2-d-arrays/80680#80680.