Pattern in magic squares

I was recently playing around with octave - a programming language for numerical computation - when I came across an interesting property in magic squares.

Magic square is a specific arrangement of unique integers in a square grid, such that sum of numbers in any row, any column or any diagonal is the same. Read more about magic squares here.

Magic squares are usually represented by matrices - square matrices. Following is an example of a 3x3 magic square:

Notice that the addition of digits across any row, any column or diagonal is same - 15 in this case.

Row 1: 8 + 1 + 6 = 15

Column 2: 1 + 5 + 9 = 15

Diagonal 1: 8 + 5 + 2 = 15

These magic square matrices are always of dimension i x i where i is an integer greater than or equal to 3.

For example, following are magic squares of dimension 5x5 and 9x9 respectively.

 

You can verify that sum of digits across any row or column or diagonal adds up to 65 - for 5x5 square and 369 for 9x9 square.

Now, consider this,

For a 3x3 magic square, we assign shades of gray colour to numbers from lightest to darkest. That is, 1 will have the lightest colour (white) and 9 will have the darkest colour (black). All the numbers in between will have the corresponding shade. With this colour assignment, the 3x3 magic square will look something like this:

The bar on the right indicates the number and its corresponding shade. Notice that this figure corresponds to the following 3x3 magic square. "1" in the middle on first row has lightest shade (white) while "9" in the middle of last row has darkest shade (black). All the other numbers have their corresponding weighted shades.

Interesting patterns emerge when we apply the same colour map for magic squares of higher dimensions.

For the illustrations below, I have considered magic squares of dimensions from 3x3 to 100x100.

I observed 3 distinct patterns for magic squares in this range of dimensions (3-100). One for all magic squares with odd number of dimension (3x3, 5x5, 7x7, 9x9 ... 99x99 ) and two for alternate even number dimensions. One for 4x4, 8x8, 12x12, 16x16, 20x20 ... so on. And another for 6x6, 10x10, 14x14, 18x18, 22x22 ... so on.

In other words,

Pattern 1 (for odd number dimensions) applies to magic squares of dimensions : 3x3, 5x5, 7x7, 9x9 ...

Pattern 2 (for 1st set of even number dimensions) applies to magic squares of dimensions : 4x4, 8x8, 12x12, 16x16, 20x20 ...

Pattern 3 (for 2nd set of even number dimensions) applies to magic squares of dimensions : 6x6, 10x10, 14x14, 18x18, 22x22 ...

The sample of patterns I found for these 3 patterns is as follows:

Pattern 1 (Odd numbers)

Notice how you can see two distinct oblique lines where dark shade lies on upper left side (indicating larger numbers) and light shade lies on lower right side (indicating smaller numbers). This figure is of a 51x51 square.

Pattern 2 (First set of even numbers) 

In the second pattern, you can see checkered distribution of numbers. That is, a group of large numbers (box of dark shade) are surrounded on four sides by groups smaller numbers (box of light shade) and vice versa. The prominence of difference between these squared boxes seem to diminish as we approach the middle horizontal line. The colours appear to converge. This figure is of a 52x52 square.

Pattern 3 (2nd set of even numbers)

In this 3rd pattern, you can see 8 distinct vertical rectangular regions. Also, there are 4 oblique lines which seem to separate darker shades (larger numbers) on top from lighter shades (smaller numbers) at bottom. But the differentiation caused by oblique lines is not as prominent as in the first pattern above. Also we can see a few small dents on left side. This is not error in data - this is part of pattern itself. This figure is of a 50x50 square.

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As a PoC, I have created the following animations which indicate how these patterns grow as the dimensions of magic square go on increasing. In each of the animation, the number in larger font at bottom of the square indicates the dimension of that magic square. The bar on the right indicates the colour scale - range of colour and their corresponding numbers.

Pattern 1 (Odd numbers)

 As the dimensions increase, (3x3, 5x5, 7x7, 9x9 ... ) the number of elements or the number of cells in the figure go on increasing but notice that the overall pattern more or less remains the same.

Pattern 2 (First set of even numbers)

As the dimensions increase (4x4, 8x8, 12x12, 16x16 ...), you can see that number of checkered boxes go on increasing, but they always seem to converge at the middle horizontal row.

Pattern 3 (2nd set of even numbers)

With the increase in dimensions (6x6, 10x10, 14x14, 18x18 ...), you can see that the four vertical rectangles as well as three oblique lines become more and more clear. The overall pattern remains the same.

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This observation was made only for magic squares upto dimension of 100x100 but I think the pattern should continue for squares of higher dimensions as well.

To see this running for yourself, run the following piece of code in Octave:

for i=3:100, 

>     imagesc(magic(i)), colormap(flipud(gray)), colorbar; 

>     xlabel(i); 

>     sleep(0.5); 

> end;

As of now, I am not aware of any mathematical/statistical implications or corollaries of these patterns of magic squares. I am yet to find any literature or resources that explain this phenomena in magic squares.

You are welcome to share below any additional findings, related resources or comments about this.