New Generalized Procedure for Magic Squares Continuation (Part IV)

Loubère and Méziriac Type Squares

A stairs

A Discussion of the New Method

Normally the Méziriac 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 Méziriac squares. In addition, the translational moves following a break may be either to the right as in A new generalized procedure (Part III) 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 Méziriac 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 to the right of the center cell of the middle 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 all rows need be moved two cells down to give the semi-magic square 6.
1
3
2
1
5
46
2
3 10
911 2
1 8
5 7
46
3
3 10 12
911 2
15 1 8
165 7 14
46 13
4
3 10 12 1921
91118 2
1517 1 8
165 7 14
46 13 20
5 Magic
3 10 12 1921
91118 25 2
151724 1 8
16235 7 14
2246 13 20
6 Semi-Magic
16 23 5 714
2246 13 20
31012 19 21
91118 25 2
151724 1 8
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 no rows or columns need to be moved for the square to be magic.
1
3 6
2
1
5
4
2
3 6
10 2
9 1
115 8
4 7
3
3 6 14
1610 13 2
912 1
115 8
154 7
4
3 17 6 14
1610 13 2
912 1 20
115 19 8
15418 7 21
5 Magic
3 17 6 2514
161024 13 2
92312 1 20
22115 19 8
15418 7 21
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 2 cells up for the square to be magic.
1
3
6 2
1
5
4
2
3 11 7
6 2
10 1
95
4 8
3
3 11 7
156 2
10 1 14
95 13
16412 8
4
3 11 720
156 19 2
1018 1 14
9175 13 21
16412 8
5
3 11 24 720
15236 19 2
221018 1 14
9175 13 21
16412 25 8
6 Magic
2210 18 114
9175 1321
16412 258
31124 720
15236 192
Method II: 4 Moves Down
  1. Place a 1 to the right of the center cell of the center 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 2 cells down but the square is not magic.
1
3
2
6 1
5
4
2
3 8
7 2
116 1
105
94
3
3 138
12 7 2
16116 1
15105
94 14
4
3 18 138
1712 7 2
16116 1 21
15105 20
94 19 14
5
3 23 18 138
221712 7 2
16116 1 21
15105 25 20
9424 19 14
6 Not Magic
9 4 24 1914
32318 13 8
221713 7 2
16116 1 21
15105 25 20
Method II: 5 Moves down
  1. Place a 1 to the right of the center cell of the center 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
29 38 47 7 9 18 27
37466 8 17 26 35
45514 16 25 34 36
41315 24 33 42 44
12 21 23 32 41 43 3
20 22 31 40 49 2 11
28 30 39 48 1 10 19
 
II Break 2 Down
4 30 14 40 17 43 27
291339 16 49 26 3
123815 48 25 2 35
372147 24 1 34 11
20 46 23 7 33 10 36
45 22 6 32 9 42 19
28 5 31 8 41 18 44
 
III Break 3 Down
12 46 31 16 1 42 27
453015 7 41 26 11
29216 40 25 10 44
20539 24 9 43 35
4 38 23 8 49 34 19
37 22 14 48 33 18 3
28 13 47 32 17 2 36
 
IV Break 4 Down
37 5 15 32 49 10 27
42131 48 9 26 36
203047 8 25 42 3
294614 24 41 2 19
45 13 23 40 1 18 35
12 22 39 7 17 34 44
28 38 6 16 33 43 11

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
3D
3D 5D
3D 5D 7D
3D5D7D9D
5D7D9D 11D
2D2D2D 2D
4D4D4D4D
1D6D6D6D
1D8D8D
1D10D
1D

This completes this section on the new generalized procedure (Part II). To return to homepage.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com