New Loubère Magic Square Method-7x7 Squares (Part IIIA)

The Complementary table Variation

A Loubere square

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.
  3. The same regular square is produced when the initial 1 is placed on the center of the first row or the center of the last column, however, this is not the case for the first column or last row.

As an example, the 7x7 regular Loubère is shown below:

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

Setting up the Complementary tables

The typical Loubère and Méziriac methods employ a consecutive number approach. i.e., the first number layed down is a 1 followed by a 2 up to the number n. The first non-consecutive approach used a number shift method where the main right diagonal is always the same, but the rest oif the numbers are consecutive but shifted. The new method employs groups of numbers, which while not consecutive as a whole are consecutive within their own group, for example (1,2,3,4,5) and (21,22,23,24,25). On this site we use this approach to manipulate the complementary table of an nxn square, generate two modified tables which are then used to construct magic squares.

As an example we begin with the complementary 7x7 table shown below:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
25
49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

Perform the next steps for a 7x7 square:

  1. Break up the table into three groups.
  2. Flip the 1st, 2nd, 3rd, 7th, 8th and 9thgroups.
  3. Move the 9th group to the first position and each of the others down one, not including lines 4, 5 and 6. These last moves are needed since the middle table does not give rise to magic squares.
  4. Label each of the groups (not including lines 4, 5 and 6) composing each complementary table. For example for a 7x7 square these labelings are IA, IIA, IB, IIB and IC and IIC.
  5. Construct the magic squares for each table using either the A, B or C configuration. The right main diagonal always uses the numbers from 22-28. For the 7x7 Loubère squares there a a total of twelve squares, six from the left hand side and six from the right hand side.
IA
 
 
 
IB
 
 
 
 
IC
 
1 2 3 4 5 6 7
49 48 47 46 45 44 43
 
8 9 10 11 12 13 14
22 23 24
25
28 27 26
42 41 40 39 38 37 36
 
15 16 17 18 19 20 21
35 34 33 32 31 30 29
7 6 5 4 3 2 1
43 44 45 46 47 48 49
 
14 13 12 11 10 9 8
22 23 24
25
28 27 26
36 37 38 39 40 41 42
 
21 20 19 18 17 16 15
29 30 31 32 33 34 35
IIA
 
 
 
IIB
 
 
 
 
IIC
 
29 30 31 32 33 34 35
7 6 5 4 3 2 1
 
43 44 45 46 47 48 49
22 23 24
25
28 27 26
14 13 12 11 10 9 8
 
36 37 38 39 40 41 42
21 20 19 18 17 16 15

The Set of Loubère Broken Diagonals

The table below correponds to the positions where the initial numerical 1's are placed. Those one's in the broken yellow diagonal correspond to by symmetry to those in the the broken light blue diagonal. This follows the rules employed in previous Loubère pages. For example see New Loubère Methods and Squares.

The set of 7x7 Broken Diagonals
1 1
11
11
1 1
1 1
1 1
1 1

The Loubère squares, which I will label Ln* (center cell#)[1D](n1,n2,n3) where (Ln* signifies a nxn Loubère square with center cell number and a break followed by 1 move Down. Since the complementary table is composed of three groups A, B and C a total of six squares are possible with the numbers n1,n2 and n3 in any of six combinations.

Examples of Magic 7x7 Squares Using the Loubère[1D] Method

Example in the order IA → IB → IC

  1. Place the first number of IA at the center of the first row of a 7x7 square (the regular historical square) and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IB and IC until the square is filled, as shown below in squares 1-2.
IA
 
 
 
IB
 
 
 
 
IC
 
1 2 3 4 5 6 7
49 48 47 46 45 44 43
 
8 9 10 11 12 13 14
22 23 24
25
28 27 26
42 41 40 39 38 37 36
 
15 16 17 18 19 20 21
35 34 33 32 31 30 29
 
1
145 12 28
7 4411 27
643 1026
5499 25
48824 4
1423 3 47
22 2 46 13
2 L7* 25 [1D](1,8,15)
371834 145 12 28
17337 4411 27 36
32643 1026 42 16
5499 2541 15 31
48824 4021 30 4
142339 2029 3 47
223819 352 46 13

Example in the order IA → IC → IB

  1. Place the first number of IA at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IC and IB until the square is filled, as shown below in squares 1-2.
IA
 
 
 
IB
 
 
 
 
IC
 
1 2 3 4 5 6 7
49 48 47 46 45 44 43
 
8 9 10 11 12 13 14
22 23 24
25
28 27 26
42 41 40 39 38 37 36
 
15 16 17 18 19 20 21
35 34 33 32 31 30 29
 
1
145 19 28
7 4418 27
643 1726
54916 25
481524 4
2123 3 47
22 2 46 20
2 L7* 25 [1D](1,15,8)
301141 145 19 28
10407 4418 27 29
39643 1726 35 9
54916 2534 8 38
481524 3314 37 4
212332 1336 3 47
223112 422 46 20

Example in the order IB → IA → IC

  1. Place the first number of IB at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IA and IC until the square is filled, as shown below in squares 1-2.
IA
 
 
 
IB
 
 
 
 
IC
 
1 2 3 4 5 6 7
49 48 47 46 45 44 43
 
8 9 10 11 12 13 14
22 23 24
25
28 27 26
42 41 40 39 38 37 36
 
15 16 17 18 19 20 21
35 34 33 32 31 30 29
 
1
838 5 28
14 374 27
1336 326
12422 25
41124 11
723 10 40
22 9 39 6
2 L7* 25 [1D](8,1,15)
441834 838 5 28
173314 374 27 43
321336 326 49 16
12422 2548 15 31
41124 4721 30 11
72346 2029 10 40
224519 359 39 6

Example in the order IB → IC → IA

  1. Place the first number of IB at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IC and IA until the square is filled, as shown below in squares 1-2.
IA
 
 
 
IB
 
 
 
 
IC
 
1 2 3 4 5 6 7
49 48 47 46 45 44 43
 
8 9 10 11 12 13 14
22 23 24
25
28 27 26
42 41 40 39 38 37 36
 
15 16 17 18 19 20 21
35 34 33 32 31 30 29
 
1
838 19 28
14 3718 27
1336 1726
124216 25
411524 11
2123 10 40
22 9 39 20
2 L7* 25 [1D](8,15,1)
30448 838 19 28
34714 3718 27 29
461336 1726 35 2
124216 2534 1 45
411524 337 44 11
212332 643 10 40
22315 499 39 20

This completes this section on New Loubère Magic Square Method (Part III). To continue this series go to New Loubère Magic Square Method (Part IIIB). To return to homepage.


Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com