New Generalized Procedure of Loubère Method (Part I)

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 7x7 regular Loubère squares is shown below as an example:

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

Normally the Loubère method involves a stepwise approach of consecutive numbers, i.e., 1,2,3... In previous methods I showed that these numbers do not have to be consecutive but may be placed semi-consecutively both frontwards or backwards as in Pendulum method I and Pendulum method II.

If the numbers are added non-consecutively,using a 7x7 example i.e. partially or fully random heptad 1,7,4,3,2,5,6 then it gets patricularly tricky keeping tracks of how the next subsequent numbers are added to complete the rest of the broken diagonals. I will show that there are two ways to accomplish this, both of which give different squares using the same heptad of opening numbers. Using the heptad above, the next number 8 can come after the 7 (the last number of the heptad) or after the 6 (the last number of the set). This method in fact may be used for any group of numbers (random or consecutive) to produce magic or semi-magic squares and may, therefore, be classified as general.

In all our examples we will be using 7x7 tables, so therefore, the heptad table is first produced by inserting in the first 7 numbers in the first row. To acquire the next row of heptad n (in this case 7) is added to all the previous numbers. This is continued until the heptad table is filled. See the first example below. Both translational and knight breaks are shown.

Construction of Generalized Procedure of (Random or Symmetrical) Loubère Square

Heptad Table I
1 4 7 62 5 3
81114 13 9 1210
151821 20 16 1917
22 25 28 27 23 2624
293235 34 30 3331
36 39 42 41 37 4038
43 46 49 48 44 4745

7x7 Squares

Method I: Group IA
  1. Generate the heptad table.
  2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  3. Move one cell right from the number 7 (the last number of the heptad).
  4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
112
3 9
513
2 14 15
11 6
78
4 10
2
112 20 25
3 9 21 22
513 18 24
2 14 15 26
111723 6
19 27 78
28 29 4 1016
3
31 37 112 20 25
413 9 21 2233
43513 18 24 3042
2 14 15 26 34 39
111723 3536 6
19 27 32 38 78
28 29 40 4 1016
4
31 37 49 112 20 25
41463 9 21 2233
43513 18 24 3042
2 14 15 26 34 3945
111723 3536 476
19 27 32 38 44 78
28 29 40 48 4 1016
Method I: Group IB
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Move one cell down from the number 7 (the last number of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
110
3 12
59
2 13
14 6
15 711
4 8
2
110 19 23
3 12 16 27
59 20 28
2 13 21 25 29
141822 6
15 24 711
26 4 817
3
34 42 110 19 23
39433 12 16 2735
59 20 28 3236
2 13 21 25 29 38
141822 31 40 6
15 24 33 37 711
26 30 41 4 817
4
34 42 46 110 19 23
39433 12 16 2735
4559 20 28 3236
2 13 21 25 29 3847
141822 31 40 446
15 24 33 37 48 711
26 30 41 49 4 817
Method II: Group IA (Non-Magic)
  1. This example is included to show that Method II: Group IA turns out to be non-magic.
  2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  3. Move one cell right from the number 3 (the last number of the set of the heptad).
  4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
111
3 8
510 15
2 12
9 6
713
4 14
2
111 21 27
3 8 18 28
510 15 25
2 12 17 22
91924 29 6
16 26 713
23 4 1420
3
34 40 111 21 27
373 8 18 2834
510 15 25 3236
2 12 17 22 32 42
91924 2939 6
16 26 31 36 713
23 30 38 43 4 820
4 Non Magic
34 40 45 111 21 27
37473 8 18 2834
44510 15 25 3236
2 12 17 22 32 4248
91924 2939 496
16 26 31 36 46 713
23 30 38 43 4 820
Method II: Group IB
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Move one cell down from the number 3 (the last number of the set of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
114
3 11
58
2 10
1215 6
79
4 13
2
114 16 24
3 11 20 2629
58 21 23
2 10 18 27
121528 6
17 25 79
22 4 1319
3
32 41 114 16 24
423 11 20 2629
58 21 23 3139
2 10 18 27 33 36
121528 3038 6
17 25 34 40 43 79
22 35 37 4 1319
4
32 41 47 114 16 24
42443 11 20 2629
4858 21 23 3139
2 10 18 27 33 3649
121528 3038 466
17 25 34 40 43 79
22 35 37 45 4 1319

Construction of Generalized Procedure of (Random or Symmetrical) Knight-Break Loubère Square

Below is again listed the heptad table I and is shown here for easier viewing. These are the only two groups that are either magic or semi-magic.

Heptad Table I
1 4 7 62 5 3
81114 13 9 1210
151821 20 16 1917
22 25 28 27 23 2624
293235 34 30 3331
36 39 42 41 37 4038
43 46 49 48 44 4745
Method I: Group IC
  1. This group of squares is semi-magic.
  2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  3. Using a knight break, move one cell up and two cells right from the number 3 (the last number of the set of the heptad).
  4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
19
3 13 15
514
2 11
8 6
710
4 12
2
19 18 26
3 13 15 23
514 17 27 29
2 11 19 28
81625 6
20 22 710
24 4 1221
3
35 38 19 18 26
403 13 15 2332
514 17 27 2937
2 11 19 28 31 4143
81625 3342 6
20 22 30 39 710
24 34 36 4 1221
4 Semi-Magic
35 38 48 19 18 26
40493 13 15 2332
46514 17 27 2937
2 11 19 28 31 4143
81625 3342 456
20 22 30 39 47 710
24 34 36 44 4 1221
Method II: Group ID
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Using a knight break, move two cells down and one cell left from the number 3 (the last number of the set of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
113
3 14
511
2 8
10 6
712
4 915
2
113 17 28
3 14 19 25
511 16 22
2 8 20 24
102126 6
18 23 29 712
27 4 915
3
33 39 113 17 28
363 14 19 2530
511 16 22 3438
2 8 20 24 35 40
102126 3237 436
18 23 29 41 712
27 31 42 4 915
4
33 39 44 113 17 28
36483 14 19 2530
49511 16 22 3438
2 8 20 24 35 4046
102126 3237 436
18 23 29 41 45 712
27 31 42 47 4 915

Results of a typical Generalized Procedure of Loubère Odd n Square (n divible by 3)

Only the Method I:Group B and Method II:Group A squares are semi magic, the rest are non-magic.

This completes this section on De La Loubère all inclusive squares (Part I). The next section deals with new Méziriac general squares (Part II). To return to homepage.


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