Magic squares

Mesosyn method for constructing singly even order (4N + 2) magic squares (e.g. 6, 10, 14, 18,····).

There are many ways to construct magic squares, but the standard (and most simple) way is to follow certain configurations/formulas which generate regular patterns. Magic squares exist for all values of n, with only one exception: it is impossible to construct a magic square of order 2 (2x2)
. Magic squares can be classified into three types: odd, doubly even (n divisible by four) and singly even (n even, but not divisible by four).
Odd magic squares are fairly easily constructed using the either the Siamese (sometimes called the de la Loubere's, or the Staircase), the Lozenge, or the de Meziriac's methods.
Doubly even magic squares are also easy to generate.
The construction of singly even magic squares is more difficult but several methods exist, including the LUX method for magic squares (due to John Horton Conway) and the Strachey method for magic squares. These established methods for creating singly even order magic squqres are not quite elegant as for odd order and doubly-even magic square in the sense that the methods are non-symmetrical and are largely empirical.

We have developed a method for constructing magic squares of singly even order in a more systemic way.

This method requires specific exchanges between a symmetrical group of four numbers:

1) a Z-shape 4-way exchange between the four symmetrical numbers.
2) a l⁄l-shape 4-way exchange between the four symmetrical numbers. l⁄l is actually an inverted N and we simply pronounce it N.
3) an — directon or I direction 2-way interchange between a pair symmetrical numbers located in the same row or in the same column.
Here we just call them the pairs and — is used to indicate the interchange between a symmetrical pair in the same row
and I is used to indicate the interchange between a symmetrical pair in the same column.
4) an '0' is used to indicate that 4 symmetrical numbers are retained in the same places without any exchange.

This method will be demonstrated by contructing a 6x6 normal magic square.

6x6 matrix 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

4-way exchange 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

4-way exchange 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

Mesosyn method for constructing a 6x6 magic square.

The sum of each row, column and diagonal is 111, the “magic number” for a 6 × 6 magic square.

First example

Step 1: All the numbers are written in the order from left to right across each row in turn, starting from the top left hand corner.
Those numbers (in brown color) on the diagonals are retained in the same place.
Enter the number of high-value numbers (19-36) in each row.
Enter the number of higher-value numbers in each column.
The higher-value numbers are the ones with higher value when compared with their symmetrical numbers in the same rows.
For examples, 6>1, 4>3, 10>9, 16>15, 24>19, 29>26 and 34>33.
In a group of 4 symmetrical numbers (for example,7,12,25, and 30), 12 and 30 are counted as higher-value numbers.
Exchange code is the sequence of exchanges you are planning to use to construct the magic square.
Whether the code will work or not can be checked in step 2, the work sheet.
Step 2: Enter the codes into their respective starting positions (Z1 or l⁄l1 positions).
To calculate the number of high-value numbers for each row, Z is counted as 1 since a Z-shape 4-way exchange will bring only one high-value numbers (19-36) into Z1 position;
l⁄l is counted as 2 since a l⁄l-shape 4-way exchange will bring two high-value numbers to the row into l⁄l1 and l⁄l3 positions;
— is counted as 0 and I is counted as 1.
To calculate the number of higher-value numbers for each column, Z is counted as 2 since it will bring two higher symmetrical value to the column into Z1 and Z3 positions;
l⁄l is counted as 1 since since it will bring only one higher symmetrical value into l⁄l1 position.
I is counted as 0 and — is counted as 1.
If the code is working, then there will be equal number of high-value numbers in each row and equal number of higher-value numbers in each column.
(You only need to check the first 3 rows and first 3 columns.)

Step 3: First exchange: a Z-type exchange between a group of 4 symmetrical numbers (2,5,32,and 35)
and a l⁄l-type exchange between a group of 4 symmetrical numbers (7,12,25,and 30)
Step 4: Second exchange: a l⁄l-type exchange between a group of 4 symmetrical numbers (3,4,33,and 34)
and a Z-type exchange between a group of 4 symmetrical numbers (13,18,19,and 24)
Step 5: Third exchange: a Z-type exchange between a group of 4 symmetrical numbers (9,10,27,and 28)
and a l⁄l-type exchange between a group of 4 symmetrical numbers (14,17,20,and 23)

Second example

Mesosyn method for constructing a 10x10 magic square.

The sum of each row, column and diagonal is 505, the “magic number” for a 10 × 10 magic square.

Mesosyn method for constructing doubly even order (4N + 4) magic squares (e.g. 8, 12, 16, 20,····).

Mesosyn method for constructing a 8 x 8 magic square.

The sum of each row, column and diagonal is 260, the “magic number” for a 8 × 8 magic square.