New Generalized Procedure for Magic Squares (Part III)

Loubère and Méziriac Type Squares

A stairs

A Discussion of the New Method

An important general principle for generating odd magic squares by the Méziriac 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 Méziriac 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 Méziriac square is shown below as an example:

3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23

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 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 IV). 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 to the Right

Method I: 1 Move to the right
  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 right.
  4. Repeat the process until the square is filled as shown below in squares 1-5.
  5. Move all columns 2 cellsto the right; this square is not magic.
1
3
2
1
5 6
4
2
3 9
8 2
1 7
5 6
410 11
3
3 9 15 16
814 2
13 1 7
5 6 12
410 11
4
3 9 15 16
81420 21 2
1319 1 7
185 6 12
410 11 17
5
3 9 15 1622
81420 21 2
131925 1 7
18245 6 12
23410 11 17
6 Not Magic
16 22 3 915
2128 14 20
1713 19 25
61218 24 5
111723 4 10
Method I: 2 Moves to the right
  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 two cells right.
  4. Repeat the process until the square is filled as shown below in squares 1-6.
  5. This square is the original Méziriac square as shown in the introduction. This square is semi-magic in that the left main diagonal is magic but the broken diagonals differ by some multiple of n.
1
3
2
1
5 6
4
2
3 9
8 2
7 1
5 6
114 10
3
3 16 9 15
8 14 2
713 1
125 6
114 10
4
3 16 9 15
20821 14 2
713 1 19
125 18 6
11417 10
5
3 16 9 2215
20821 14 2
72513 1 19
24125 18 6
11417 10 23
Method I: 3 Moves to the right
  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 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 2 spaces to the left for the square to be magic.
1
3
2
1
65
4
2
3 9
8 2
7 1
65
411 10
3
3 15 916
148 2
7 1 13
65 12
411 10
4
3 16 916
14218 20 2
719 1 13
6185 12
17411 10
5
3 16 22 916
14218 20 2
25719 1 13
6185 12 24
17411 23 10
6 (Magic)
22 9 16 315
8202 14 21
19113 25 7
51224 6 18
112310 17 4
Method I: 4 Moves to the right
  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 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 right for the square to be magic.
1
3
2
1
65
4
2
3 9
8 2
7 1
65
104 11
3
3 16 159
14 8 2
137 1
1265
104 11
4
3 16 159
212014 8 2
19137 1
1265 18
104 17 11
5
3 22 16 159
212014 8 2
19137 1 25
1265 24 18
10423 17 11
6 (Magic)
9 3 22 1615
22120 14 8
251913 7 1
18126 5 24
11104 23 17
Method I: 5 Moves to the right
  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 five cells right. The next number 6 would land directly over the 5, so that this square is not magic.
Method I: Three 7x7 fully Configured Magic Squares with a Typical Break Move

I Break 3 Right
12 45 29 20 4 37 28
443519 3 36 27 11
34182 42 26 10 43
17141 25 9 49 33
7 40 24 8 48 32 16
39 23 14 47 31 15 6
22 13 46 30 21 5 38
 
II Break 4 Right
37 4 20 29 45 12 28
31935 44 11 27 36
183443 10 26 42 2
33499 25 41 1 17
48 8 24 40 7 16 32
14 23 39 6 15 31 47
22 38 5 21 30 46 13
 
III Break 5 Right
45 20 37 12 29 4 28
193611 35 3 27 44
421034 2 26 43 18
9331 25 49 17 41
32 7 24 48 16 40 8
6 23 47 15 39 14 31
22 46 21 38 13 30 5

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 three 7x7 examples are shown in white in the following table:

Size
5x5
7x7
9x9
11x11
 
Squares that Break to the Right
1R 3R 5R 2R4R
1R3R5R 7R 2R4R 6R
1R3R 5R7R 9R 2R4R 6R 8R
1R3R5R7R9R 11R 2R4R6R8R 10R

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


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