New Generalized Procedure for Magic Squares Continuation (Part II)

Loubère and Méziriac Type Squares

A stairs

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
  1. Place a 1 in the center cell of the first row.
  2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
  3. Move one cell down.
  4. Repeat the process until the square is filled as shown below in squares 1-5.
  5. As shown below no column or row need be moved and the square is the known regular magic shown at the beginning.
1
1
5
46
3
2
2
1 8
57
46
10 3
11 2 9
3
1 815
57 14 16
4613
1012 3
11 2 9
4
17 1 815
57 14 16
4613 20
101219 21 3
1118 2 9
5 Magic
17 24 1 815
2357 14 16
4613 20 22
101219 21 3
111825 2 9
Method II: 2 Moves Down
  1. Place a 1 in the center cell of the first row.
  2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
  3. Move two cells down.
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. As shown below all rows need to be moved 2 spaces up for the square to be magic.
1
1
5
4
6 3
2
2
1 9
115 8
47
6 3
10 2
3
12 1 9
115 8
47 15
6 14 3
1013 2 16
4
12 1 209
11519 8
4187 21 15
176 14 3
1013 2 16
5
23 12 1 209
11519 8 22
4187 21 15
17625 14 3
102413 2 16
6 Magic
4 18 7 2115
17625 14 3
102413 2 16
23121 20 9
11519 8 22
Method II: 3 Moves Down
  1. Place a 1 in the center cell of the first row.
  2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
  3. Move three cells down.
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. As shown below all rows need to be moved 1 space down for the square to be magic.
1
1
5
4
3
6 2
2
10 1
5 9
4 8
117 3
6 2
3
10 1 14
513 9
412 8 16
117 3
6 2 15
4
10 18 1 14
17513 21 9
412 8 16
117 20 3
619 2 15
5
10 18 1 1422
17513 21 9
41225 8 16
11247 20 3
23619 2 15
6 Magic
23 6 19 215
10181 14 22
17513 21 9
41225 8 16
11247 20 3
Method II: 4 Moves Down
  1. Place a 1 in the center cell of the first row.
  2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
  3. Move four cells down.
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. As shown below all rows need to be moved 1 space up but the square is not magic.
1
6 1
5
4
3
2
2
11 6 1
105
4 9
8 3
7 2
3
11 6 1 16
105 15
4 14 9
13 8 3
127 2
4
11 6 1 2116
105 20 15
419 14 9
1813 8 3
17127 2
5
11 6 1 2116
10525 20 15
42419 14 9
231813 8 3
17127 2 22
6 Not Magic
10 5 25 2015
42419 14 9
231813 8 3
17127 2 22
1161 21 16
Method II: 5 Moves down
  1. Place a 1 in the center cell of the first row.
  2. Fill in the numbers in a stepwise manner, until blocked by a previous number.
  3. 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
38477 9 18 27 29
4668 17 26 35 37
51416 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
301440 17 43 27 4
133916 49 26 3 29
381548 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
463116 1 42 27 12
30157 41 26 11 45
21640 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
51532 49 10 27 37
213148 9 26 36 4
30478 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 semi-magic squares are in light green color, while the non-magic squares are in orange color. Squares formed from n div by 3 forms only one magic square in the semi-magic set; the other n - 1 are semi-magic squares. In addition, the four 7x7 examples are shown in white in the following table:

Squares That Break Down
5x57x79x911x11
1D
1D 3D
1D 3D 5D
1D3D5D7D
3D5D7D 9D
5D7D9D 11D
2D2D2D2D
4D4D4D4D
6D6D6D
8D8D
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. E-Mail: Fiboguti89@Yahoo.com