Magic Squares of odd order

Magic Squares of odd order by Marios Mamzeris
 

The following is a method of creating any odd-order associative magic square. I had the idea of this technique many years ago, back in 1988, and I finally decided to show it in public (all rights reserved).

In brief, a Magic Square is a n x n square grid (where n is the number of cells on each side) filled with distinct positive integers in the range 1,2...,n² such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal.

An associative magic square is a magic square with a further property that every number added to the number equidistant, in a straight line, from the center gives n² + 1. They are also called symmetric magic squares. All associated magic square is self-complementary magic squares as well. (Wikipedia)

​To create any odd-order associative magic square quickly and hassle-free, we just need to make two simple passes, that I describe as table transformations (shifts). The following three tables are depicting the creation of a 31x31 magic square.

1) We first get the full table filled with all numbers from 1 to n² as we go (1^st table below showing the cells shifting in green color). Then we make a left shift of all rows, skipping the middle one, starting with moving one number (cell) in the first row while moving an extra more number (cell) in each next row to n-1 numbers in the last (n^th ) row.

2) The second action is similar to the 1^st, but instead of shifting cells in rows, we now shift cells in columns. It starts with moving one number (cell) upwards from the left side while moving an extra more number (cell) in each next column to n-1 numbers in the last (n^th ) column, skipping the middle column this time. That is the 2^nd table below, showing the cells shifting in blue color)

That’s it. A magic square is ready. That is the 3^rd table below, with light and dark grey cells and yellow diagonals.

1. Initial table with all the numbers from 1 to n² in sequence (The green arrow shows the direction of the cell shift that will follow and the green cells are the ones that will shift.)

2. First transformation (primary cell shift). This table is the result of the shift depicted in green color above. (The blue arrow shows the direction of the 2^nd and final cell shift and the blue cells are the ones that will shift.)

3. Final transformation (secondary cell shift) that gives the Magic Square.

An alternative way (based on the above shift pattern) is to create a magic square by performing the cell shifting in opposite directions for each half of the table, always skipping the middle row or column.

Initial square with numbers 1 - n² in sequence

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49

After the first pass. Notice how each column is equal to Σ

	2	3	4	5	6	7	1
	10	11	12	13	14	8	9
	18	19	20	21	15	16	17
	22	23	24	25	26	27	28
	33	34	35	29	30	31	32
	41	42	36	37	38	39	40
	49	43	44	45	46	47	48

Second pass completed and now each row is also equal to Σ. The Magic Square is ready

Some Notes

Below are some simple notes to help the reader understand what is happening to the sums of rows and columns of the initial (1) table n x n, and how each pass (series of shifts) affects them, and leads to the construction of a symmetrical magic square.

n = square side size (magic square is n x n in size)
M = (n²+1)/2 (middle number, in the center)
Σ = Mn (Sum of each row, column, diagonal of the magic square) 
r = the number of rows or columns away from the middle.

As we create the initial square (1) with all the numbers from 1 to n² in sequence, we notice the ‘weight’ difference in each row and column, compared to the middle row and column accordingly. For columns, going left from the middle one, the column’s sum is Σ - rn. For columns going to the right from the middle one, the sum is Σ + rn. For rows, going up from the middle row, the sum is  Σ - (rn²). For rows, going down from the middle row, the sum is  Σ + (rn²).

The first pass, shifting cells in rows to the left, starting with one cell (1^st top left corner) and incrementing one more cell in every next row, skipping the middle one, gives us a square (2) where all the row sums are equal to Σ. Notice how the left row has gained + 1 to the first cell on the top, adding one extra in every next cell going to the last cell on the bottom, skipping the middle one. Each next row is following a decremental pattern of gains, and they all affect the row sums in a way that compensates the initial (1) sums to equalize with the middle row.

In a similar fashion, we observe the same behavior in column sums. While they remain as detailed in the first paragraph above, in squares (1) and (2), when we perform the equivalent cell shifting in columns as we previously did in rows, we notice how the column sums are compensated and become equal to Σ.

Below we can see:

On the left side, the squares with each cell value in relation to n.

On the right side, the square progress with actual values. Square (3) is the final and magic one.

In the green-colored cells, the sums of each row and column. The diagonals are always equal to Σ.

In the blue-colored cells, are the initial middle row and column cells, and their location throughout the passes.

								Σ										175		Σ
n-6	n-5	n-4	n-3	n-2	n-1	n		Σ-(3n2)		1	2	3	4	5	6	7		28		Σ-(3n2)
n+1	n+2	n+3	n+4	n+5	n+6	n+7		Σ-(2n2)		8	9	10	11	12	13	14		77		Σ-(2n2)
n+8	n+9	n+10	n+11	n+12	n+13	n+14		Σ-(1n2)		15	16	17	18	19	20	21		126		Σ-(1n2)
n+15	n+16	n+17	n+18	n+19	n+20	n+21		Σ		22	23	24	25	26	27	28		175		Σ
n+22	n+23	n+24	n+25	n+26	n+27	n+28		Σ+(1n2)		29	30	31	32	33	34	35		224		Σ+(1n2)
n+29	n+30	n+31	n+32	n+33	n+34	n+35		Σ+(2n2)		36	37	38	39	40	41	42		273		Σ+(2n2)
n+36	n+37	n+38	n+39	n+40	n+41	n+42		Σ+(3n2)		43	44	45	46	47	48	49		322		Σ+(3n2)
								Σ										175		Σ
										154	161	168	175	182	189	196
Σ-(3n)	Σ-(2n)	Σ-(1n)	Σ	Σ+(1n)	Σ+(2n)	Σ+(3n)				Σ-(3n)	Σ-(2n)	Σ-(1n)	Σ	Σ+(1n)	Σ+(2n)	Σ+(3n)
								Σ										175		Σ
n-5	n-4	n-3	n-2	n-1	n	n-6		Σ-(3n2)		2	3	4	5	6	7	1		28		Σ-(3n2)
n+3	n+4	n+5	n+6	n+7	n+1	n+2		Σ-(2n2)		10	11	12	13	14	8	9		77		Σ-(2n2)
n+11	n+12	n+13	n+14	n+8	n+9	n+10		Σ-(1n2)		18	19	20	21	15	16	17		126		Σ-(1n2)
n+15	n+16	n+17	n+18	n+19	n+20	n+21		Σ		22	23	24	25	26	27	28		175		Σ
n+26	n+27	n+28	n+22	n+23	n+24	n+25		Σ+(1n2)		33	34	35	29	30	31	32		224		Σ+(1n2)
n+34	n+35	n+29	n+30	n+31	n+32	n+33		Σ+(2n2)		41	42	36	37	38	39	40		273		Σ+(2n2)
n+42	n+36	n+37	n+38	n+39	n+40	n+41		Σ+(3n2)		49	43	44	45	46	47	48		322		Σ+(3n2)
								Σ										175		Σ
										175	175	175	175	175	175	175
Σ	Σ	Σ	Σ	Σ	Σ	Σ				Σ	Σ	Σ	Σ	Σ	Σ	Σ
								Σ										175		Σ
n+3	n+12	n+17	n-2	n+23	n+32	n+41		Σ		10	19	24	5	30	39	48		175		Σ
n+11	n+16	n+28	n+6	n+31	n+40	n-6		Σ		18	23	35	13	38	47	1		175		Σ
n+15	n+27	n+29	n+14	n+39	n	n+2		Σ		22	34	36	21	46	7	9		175		Σ
n+26	n+35	n+37	n+18	n-1	n+1	n+10		Σ		33	42	44	25	6	8	17		175		Σ
n+34	n+36	n-3	n+22	n+7	n+9	n+21		Σ		41	43	4	29	14	16	28		175		Σ
n+42	n-4	n+5	n+30	n+8	n+20	n+25		Σ		49	3	12	37	15	27	32		175		Σ
n-5	n+4	n+13	n+38	n+19	n+24	n+33		Σ		2	11	20	45	26	31	40		175		Σ
								Σ										175		Σ
										175	175	175	175	175	175	175
Σ	Σ	Σ	Σ	Σ	Σ	Σ				Σ	Σ	Σ	Σ	Σ	Σ	Σ