The constant sum in every row, column and diagonal is called the magic constant or magic sum or magic number, M.
The magic constant of a normal magic square depends only on n and has the value M = n(n² + 1)/2
For normal magic squares of order n = 3, 4, 5, 6, 7,8,····, the magic constants are:
15, 34, 65, 111, 175, 260,····.
Method for constructing a magic square of odd order (e.g. 3, 5, 7, 9,····).
A method for constructing magic squares of odd order was published by the French diplomat de la Loubère in his book A new historical relation of the kingdom of Siam (Du Royaume de Siam, 1693), under the chapter entitled The problem of the magical square according to the Indians. De la Loubère's method (Siamese method) operates as follows:
Starting from the central column of the first row with the number 1, the fundamental movement for filling the squares is diagonally up and right, one step at a time. If a filled square is encountered, one moves vertically down one square instead, then continuing as before. When a move would leave the square, it is wrapped around to the last row or first column, respectively.
Order-3 magic squares.
The sum of each row, column and diagonal is 15, the magic number for a 3 × 3 magic square.
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Source: Wikipedia's Siamese method
A simple example of the Siamese method. Starting with "1" in the middle colomn of the top row. Boxes are filled diagonally up and right until you are blocked. At this point you drop down one square and then continuing as before. When a move would leave the square, it is wrapped around to the last row or first column, respectively.
Order-5 magic squares.
The sum of each row, column and diagonal is 65, the magic number for a 5 × 5 magic square.
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Bachet de Méziriac method.
You can also construct a magic square of odd order by starting with "1" in the middle colomn immediately above the the center square. The method is the same as above except that when blocked the next number is placed in the same column two rows up rather than one row down.
Reference: William H.Benson and Oswald Jacoby (1976) New Recreations With Magic Squares. Published by General Publishing Company, Ltd., Toronto, Ontario, Canada.
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Method for constructing a doubly even order (4N + 0) magic square (e.g. 4, 8, 12, 16,····).
A method of constructing a magic square of doubly even order. Doubly even means an even multiple of an even integer; or 4N (e.g. 4, 8, 12,····), where N is an integer (e.g. 1, 2. 3,····). The resulting square was also called a mystic square by Joel B. Wolowelsky and David Shakow in their article describing a method for constructing a magic square whose order is a multiple of 4.
Method for constructing a 4 x 4 magic square.
A 4 × 4 magic square can be constructed by writing out the numbers from 1 to 16 consecutively in a 4 × 4 matrix and then interchanging those numbers on the diagonals that are equidistant from the center. (step 2→step 3).
Order-4 magic squares.
The sum of each row, column and diagonal is 34, the magic number for a 4 × 4 magic square.
The sum of each row, column and diagonal is 260, the magic number for a 8 × 8 magic square.
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.
Step 2a: Interchanging those numbers (in red color) on the diagonals that are equidistant from the center. (Same method for constructing a 4 x 4 magic square.)
Step 2b: Generate a 4x4 sub-square lines. Note that 16 numbers are crossed by the lines. These numbers (in blue color) will be interchanged.
Step 3: These numbers (in blue color from step 2) can be interchanged with their diametrically opposite numbers to create a magic square:
(25<->40, 33<->32), (4<->61, 5<->60), (18<->47, 11<->54), (14<->51, 23<->42).
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Simplified method to create doubly even magic squares of higher order.
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.
Step 2: (a)Generate two major diagonal lines. Note that 16 numbers (in red color) are crossed by these 2 lines in 8x8 magic squares, 24 numbers in 12x12 magic squares and 32 numbers in 16x16 magic squares.
(b)Generate 4x4(N-1) sub-square lines. Note that 16 numbers (in blue color) are crossed by these lines in 8x8 magic squares, 48 numbers in 12x12 magic squares and 96 numbers in 16x16 magic squares.
Step 3:
Write down these numbers in the order from left to right across each row. Their numbers will be exactly half of the total squares, e.g. 32 numbers in 8x8 magic squares, 72 numbers in 12x12 magic squares and 128 numbers in 16x16 magic squares.
In creating a 8x8 magic squares these 32 numbers are 1,4,5,8,10,11,14,15,18,19,22,23,25,28,29,32,33,36,37,40,42,43,46,47,50,51,54,55,57,60,61,64.
All we need to do is to fill these squares with the same numbers in reverse order:
64,61,60,57,55,54,51,50,47,46,43,42,40,37,36,33,32,29,28,25,23,22,19,18,15,14,11,10,8,5,4,1.
Order-12 magic squares.
The sum of each row, column and diagonal is 870, the magic number for a 12 × 12 magic square.
In creating 12x12 magic squares, the 72 numbers that need to be reversed are 1,4,5,8,9,12,14,15,18,19,22,23,26,27,30,31,34,35,37,40,41,44,45,48,49,52,53,56,57,60,62,63,66,67,70,71,
74,75,78,79,82,83,85,88,89,92,93,96,97,100,101,104,105,108,110,111,114,115,118,119,122,123,126,127,130,131,133,136,137,140,141,144.
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Method for constructing a singly even order (4N + 2) magic square (e.g. 6, 10, 14, 18,····).
It is more difficult to construct a magic square of singly even order. Singly even means an even integers, where they are divisible by 2, but not by 4.
The 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-symmtrical and are largely empirical. We have developed a method for constructing magic squares of singly even order in a more systemic and symmetrical way. You can
click here to find out.
Order-6 magic squares ( Constructed with Strachey's method ).
The sum of each row, column and diagonal is 111, the magic number for a 6 × 6 magic square.
Step 1: Write out the numbers from 1 to 36 consecutively in a 6 × 6 matrix.
Step 2: Divide the grid into 4 quarters each having 9 cells and name them crosswise thus
A C
D B
Fill up the subsquare A with the numbers 1 to 9, then the subsquare B with the numbers 10 to 18, then the subsquare C with the numbers 19 to 27, then the subsquare D with the numbers 28 to 36.
Step3: Using the Siamese method (De la Loubère method) for order-3 magic squares and complete the individual magic subsquares A,B,C,D,
Step 4: A magic square can then be constructed by exchanging just 3 pairs of numbers ( 8<->35, 4<->31, and 5<->32 ).
These pairs are located in corresponding cells between subsquare A and subsquare D.
6x6 matrix
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Order-10 magic squares.
The sum of each row, column and diagonal is 505, the magic number for a 10 × 10 magic square.
Strachey's method can be used to construct singly even order (4N + 2) magic squares(e.g. 6, 10, 14, 18,····).
The number of corresponding pairs that have to be exchanged is ( n²/4 )-n .
A 6x6 magic magic square (N=1, n=6) can be made by additional exchanging 3 pairs of corresponding numbers.
A 10x10 magic square (N=2,n=10) can therefore be made by additional exchanging 15 pairs of corresponding numbers (in yellow color).
However, the pattern of exchanging seems to be quite arbitrary and non-symmetrical.
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We have developed a method for constructing magic squares of singly even order in a more systemic way. Click here to find out.
