New Generalized Procedure for Magic Squares (Part I)

Loubère and Méziriac Type Squares

A stairs

A Discussion of the New Method

An important general principle for generating odd magic squares by the De La Loubère method is that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n2 + 1). The properties of these regular or associated Loubère squares are:

  1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
  2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n2 + 1, i.e., or twice the number in the center cell and are complementary to each other.

The 5x5 regular Loubère square is shown below as an example:

17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9

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 on this page or down as in the next page (continuation) (Part II). 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 Complete Loubère Squares

A Break to the Right

Method I: 1 Move to the right
  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 right.
  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 not magic.
1
1
56
4
3
2
2
1 7
56
41011
9 3
2 8
3
1 713
56 12
41011
91516 3
14 2 8
4
19 1 713
56 12 18
41011 17
91516 3
142021 2 8
5 Not Magic
19 25 1 713
2456 12 18
41011 17 23
91516 22 3
142021 2 8
Method I: 2 Moves to the right
  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 right.
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. As shown below all columns need to be moved 2 spaces to the left for the square to be magic and the square is semi-magic.
1
1
5 6
4
3
2
2
1 7
5 6
410 11
9 3
8 2
3
13 1 7
125 6
410 11
169 15 3
814 2
4
13 1 197
12518 6
41710 11
169 15 3
82114 2 20
5
25 13 1 197
12518 6 24
41710 23 11
16922 15 3
82114 2 20
6 (Semi Magic)
1 19 7 2513
18624 12 5
102311 4 17
22153 16 9
14220 8 21
Method I: 3 Moves to the right
  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 right.
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. As shown below all columns need to be moved 1 space to the right for the square to be magic and the square is magic.
1
1
5 6
4
3
2
2
7 1
5 6
411 10
9 3
8 2
3
7 1 13
512 6
411 10
159 16 3
8 2 14
4
7 19 1 13
18512 6
411 10 17
159 16 3
21820 2 14
5
7 19 1 1325
18512 24 6
41123 10 17
15229 16 3
21820 2 14
6 (Magic)
25 7 19 113
6185 12 24
17411 23 10
31522 9 16
14218 20 2
Method I: 4 Moves to the right
  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 right; same as one cell left .
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. As shown below all columns need to be moved 1 space to the left for the square to be magic and the square is magic.
1
1
65
4
3
2
2
7 1
65
4 11 10
9 3
8 2
3
13 7 1
65 12
4 11 10
1615 9 3
148 2
4
13 7 1 19
65 18 12
417 11 10
1615 9 3
20148 2 21
5
13 7 1 2519
6524 18 12
42317 11 10
221615 9 3
20148 2 21
6 (Magic)
7 1 25 1913
62418 12 6
231711 10 4
16159 3 22
1462 21 20
Method I: 5 Moves to the right
  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 right. 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 3 Right
9 49 33 17 1 41 25
483216 7 40 24 8
31156 39 23 14 47
21538 22 13 46 30
4 37 28 12 45 29 20
36 27 11 44 35 19 3
26 10 43 34 18 2 42
 
II Break 4 Right
41 1 17 33 49 9 25
71632 48 8 24 40
153147 14 23 39 6
304613 22 38 5 21
45 12 28 37 4 20 29
11 27 36 3 19 35 44
26 42 2 18 34 43 10
 
III Break 5 Right
49 17 41 9 33 1 25
16408 32 7 24 48
391431 6 23 47 15
13305 22 46 21 38
29 4 28 45 20 37 12
3 27 44 19 36 11 35
26 43 18 42 10 34 2
 
IV Break 6 Right
17 9 1 49 41 33 25
8748 40 32 24 16
64739 31 23 15 14
463830 22 21 13 5
37 29 28 20 12 4 45
35 27 19 11 3 44 36
26 18 10 2 43 42 34

Summary of Generalized Procedure for Right Break 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: R(right). 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:

Size
5x5
7x7
9x9
11x11
 
Squares that Break to the Right
2R 4R 1R 3R5R
2R4R6R 1R 3R5R 7R
2R4R 6R8R 1R 3R5R 7R 9R
2R4R6R8R10R 1R 3R5R7R9R11R

This completes this section on the new generalized procedure (Part I). The next section deals with Continuation of a new generalized procedure (Part II). To return to homepage.


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