New Generalized Procedure for Magic Squares Continuation (Part II)
Loubère and Méziriac Type Squares
A Discussion of the New Method
Normally the Loubère method involves a stepwise approach of consecutive numbers, i.e., 1,2,3...
In this report the initial number 1 does not have to be placed at the middle of the first row cell but may be placed in any available position on the first row or
the center column to give modified Loubère squares. In addition, the translational moves following a break may be either
to the right as in A new generalized procedure (Part I) or down as on this page. For simplicity
5x5 squares will be used for demonstration. Fully formed 7x7 examples will be shown at the end.
Using first principles we use one configuration of squares to generate all the other configurations in the set. For example, the first step
involves always placing the initial number 1 onto the middle cell of the first row and doing the stepwise
Loubère addition of numbers onto the square. If the square (in this case a 5x5) has the right number set 11,12,13,14,15 on the right main diagonal nothing has
to be done. If these numbers are not on the main right diagonal then column or rows must be moved an m number of moves, where
m is 1..n until the square is in the right configuration. This new configuration will be shown to produce both Loubère and Méziriac
type Squares.
Construction of Generalized Procedure for Loubère and Méziriac Squares
A Break Vertically Down
Method II: 1 Move Down
 Place a 1 in the center cell of the first row.
 Fill in the numbers in a stepwise manner, until blocked by a previous number.
 Move one cell down.
 Repeat the process until the square is filled as shown below in squares 15.
 As shown below no column or row need be moved and the square is the known regular magic shown at the beginning.

⇒ 

⇒ 
3
  1 
8  15 
 5  7 
14  16 
4  6  13 
 
10  12  
 3 
11   
2  9 

⇒ 
4
17   1 
8  15 
 5  7 
14  16 
4  6  13 
20  
10  12  19 
21  3 
11  18  
2  9 

⇒ 
5 Magic
17  24  1 
8  15 
23  5  7 
14  16 
4  6  13 
20  22 
10  12  19 
21  3 
11  18  25 
2  9 

Method II: 2 Moves Down
 Place a 1 in the center cell of the first row.
 Fill in the numbers in a stepwise manner, until blocked by a previous number.
 Move two cells down.
 Repeat the process until the square is filled as shown below in squares 16.
 As shown below all rows need to be moved 2 spaces up for the square to be magic.

⇒ 

⇒ 
3
 12  1 
 9 
11  5  
8  
4   7 
 15 
 6  
14  3 
10   13 
2  16 

⇒ 
4
 12  1 
20  9 
11  5  19 
8  
4  18  7 
21  15 
17  6  
14  3 
10   13 
2  16 

⇒ 
5
23  12  1 
20  9 
11  5  19 
8  22 
4  18  7 
21  15 
17  6  25 
14  3 
10  24  13 
2  16 

⇒ 
6 Magic
4  18  7 
21  15 
17  6  25 
14  3 
10  24  13 
2  16 
23  12  1 
20  9 
11  5  19 
8  22 

Method II: 3 Moves Down
 Place a 1 in the center cell of the first row.
 Fill in the numbers in a stepwise manner, until blocked by a previous number.
 Move three cells down.
 Repeat the process until the square is filled as shown below in squares 16.
 As shown below all rows need to be moved 1 space down for the square to be magic.

⇒ 

⇒ 
3
10   1 
14  
 5  13 
 9 
4  12  
8  16 
11   7 
 3 
 6  
2  15 

⇒ 
4
10  18  1 
14  
17  5  13 
21  9 
4  12  
8  16 
11   7 
20  3 
 6  19 
2  15 

⇒ 
5
10  18  1 
14  22 
17  5  13 
21  9 
4  12  25 
8  16 
11  24  7 
20  3 
23  6  19 
2  15 

⇒ 
6 Magic
23  6  19 
2  15 
10  18  1 
14  22 
17  5  13 
21  9 
4  12  25 
8  16 
11  24  7 
20  3 

Method II: 4 Moves Down
 Place a 1 in the center cell of the first row.
 Fill in the numbers in a stepwise manner, until blocked by a previous number.
 Move four cells down.
 Repeat the process until the square is filled as shown below in squares 16.
 As shown below all rows need to be moved 1 space up but the square is not magic.

⇒ 

⇒ 
3
11  6  1 
 16 
10  5  
 15 
4   
14  9 
  13 
8  3 
 12  7 
2  

⇒ 
4
11  6  1 
21  16 
10  5  
20  15 
4   19 
14  9 
 18  13 
8  3 
17  12  7 
2  

⇒ 
5
11  6  1 
21  16 
10  5  25 
20  15 
4  24  19 
14  9 
23  18  13 
8  3 
17  12  7 
2  22 

⇒ 
6 Not Magic
10  5  25 
20  15 
4  24  19 
14  9 
23  18  13 
8  3 
17  12  7 
2  22 
11  6  1 
21  16 

Method II: 5 Moves down
 Place a 1 in the center cell of the first row.
 Fill in the numbers in a stepwise manner, until blocked by a previous number.
 Move five cells down. The next number 6 would land directly over the 5, so that this square is not magic.
Method I: Four 7x7 fully Configured Magic Squares with a Typical Break Move
I Break 1 Down
30  39  48 
1  10 
19  28 
38  47  7 
9  18 
27  29 
46  6  8 
17  26 
35  37 
5  14  16 
25  34 
36  45 
13  15  24 
33  42 
44  4 
21  23  32 
41  43 
3  12 
22  31  40 
49  2 
11  20 


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


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


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

Summary of Generalized Procedure for Break Down Squares
This table summarizes square sizes 5x5 to 11x11 for the generalized procedure above. Each of the cells corresponds to where the initial number 1 would go along with
the number of moves after the break: D(down). The semimagic squares are in light green color,
while the nonmagic squares are in orange color. Squares formed
from n div by 3 forms only one magic square in the semimagic set; the other n  1 are semimagic squares. In addition,
the four 7x7 examples are shown in white in the following table:
Squares That Break Down
5x5  7x7  9x9  11x11 
 
 1D 
1D  3D 
1D  3D  5D 
1D  3D  5D  7D 
3D  5D  7D 
9D 
5D  7D  9D 
11D 
2D  2D  2D  2D 
4D  4D  4D  4D 
 6D  6D  6D 
 8D  8D 
 10D 
This completes this section on the new generalized procedure (Part II). The next section deals with
A new generalized procedure (Part III) starting with the Méziriac procedure. To return to homepage.
Copyright © 2009 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com