New De La Loubère Method and Squares (Part I)

Regular and Non-Regular Loubère Squares

The Full Monty I

A Loubere square

A Discussion of the New Methods

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.

The 5x5 and 7x7 regular Loubère squares are shown below as examples:

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

This web page has been modified from the previous to take into account new findings because the previous Loubère method has been found to be a subset of a general method. It will be shown that the initial numeral 1 can be placed anywhere on two broken diagonals, in which the original Loubère method is placed on the top diagonal and the the second initial 1 is placed on the bottom diagonal. This page will show that all the squares having an initial 1 on the broken diagonal are related to one another. It will also be shown that both regular and non-regular squares are produced. In fact, more non-regular squares are generated than regular squares. This goes for this and other Loubère methods that I will be discussing. In addition, a new Loubère-Knight method method is also part of this method.

All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the light grey main diagonal behave differently from typical Loubère squares. After a break/1 move down for the broken yellow diagonal and break/1 move left for the broken light blue diagonal, one regular square and n - 1 non-regular squares are produced. Squares belonging to one diagonal group are identical to another square on the other diagonal group. Using the 7x7 square as an example shows the two diagonals and typical 1 positions. The second table shows the equation and value of the center cell of each square (starting with the square generated from 1 in the first row) where the values range from ½(n2 - n + 2) to ½(n2 + 7) for squares breaking to the right or to the bottom.

1 1
11
11
1 1
1 1
1 1
1 1
 
center Value
EquationValue LeftEquationValue Down
½(n2 - 5)22½(n2 + 1)25
½(n2 - 3)23½(n2 + 3)26
½(n2 - 1)24½(n2 + 5)27
½(n2 + 1)25½(n2 + 7)28
½(n2 + 3)26½(n2 - 5)22
½(n2 + 5)27½(n2 - 3)23
½(n2 + 7)28½(n2 - 1)24

These new Loubère squares, which I will label Ln* (center cell#) (DOWN or LEFT) where (Ln* signifies a nxn Loubère square with the center cell number of the square and breaking either down or to the left. Thgis would make the original Loubère squares depicted in the introduction as L5* 13 DOWN and L7* 25 DOWN. The squares exhibit the following properties:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. Odd squares divisible by 3, i.e., 3(2n + 1) obey a simple modified rule. These square may be magic or semi-magic and the sum of the left main diagonal cycles through the triad S, S + n and S - n. For example, for a 9x9 square there are three cycles of 369, 370 and 369.

Construction of regular and non-regular Loubère squares

5x5 Squares

  1. To generate the non-regular square, L5* 15 LEFT, place a 1 into the center of the last row of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move one cell to the left.
  3. Repeat the process until the non-regular square is filled, as shown below in squares 1-5.
  4. Note that square 5L, L5* 15 LEFT, is identical (rotated along the right main diagonal) to square 5D, L5* 15 DOWN, constructed from the yellow diagonal.
1
6 5
4
3
2
1
2
6 5
4 11 10
9 3
8 2
7 1
3
6 5 12
4 11 10
1615 9 3
148 2
13 7 1
4
6 5 18 12
417 11 10
1615 9 3
20148 2 21
13 7 1 19
5L
6 5 24 18 12
42317 11 10
221615 9 3
20148 2 21
13 7 1 25 19
5D
19 21 3 10 12
2529 11 18
1815 17 24
71416 23 5
13 20 22 4 6
A five series

Note that two pairs sum to 30 including the center square when multiplied by 2 and ten pairs sum to 25. In addition none of these pairs are complementary, as in the regular Loubère squares, as shown in the connectivities of the complementary table.

7x7 Squares

The construction of the 7x7 non-regular Loubère square L7* 28 LEFT from the broken light blue diagonal set follows the same line.

1
8 7
146 15
5 13
12 4
11 3
10 2
9 1
2
16 8 7 24
146 23 15
5 22 21 13
29 28 20 12 4
27 19 11 3
26 18 10 2
25 17 9 1
3
16 8 7 48 40 32 24
14647 39 31 23 15
54638 30 22 21 13
45 3729 28 20 12 4
36 35 27 19 11 3 44
34 26 18 10 2 43 42
25 17 9 1 49 41 33
A seven series

Note that three pairs sum to 56 including the center square when multiplied by 2 and twenty one pairs sum to 49. In addition none of these pairs are complementary, as in the regular Loubère squares, as shown in the connectivities of the complementary table.

Examples of 9x9 regular magic and non-regular Semi- Magic Loubère Squares

However, the use of odd squares divisible by three, as mention previously recycles through a triad of magic and semi-magic squares. The following three 9x9 squares show that the sum of the left diagonal is equal to 369, 378 and 360, and this is followed by triad recycling.

Magic I L9* 41 DOWN
47 58 69 80 1 12 23 34 45
576879 9 11 22 33 44 46
67788 10 21 32 43 54 56
77718 20 31 42 53 55 66
6 17 19 30 41 52 63 65 76
16 27 29 40 51 62 64 75 5
26 28 39 50 61 72 74 4 15
36 38 49 60 71 73 3 14 25
37 48 59 70 81 2 13 24 35
 
Semi-magic II L9* 42 DOWN
48 59 70 81 2 13 24 35 37
586980 1 12 23 34 45 47
68799 11 22 33 44 46 57
78810 21 32 43 54 56 67
7 18 20 31 42 53 55 66 77
17 19 30 41 52 63 65 76 6
27 29 40 51 62 64 75 5 16
28 39 50 61 72 74 4 15 26
38 49 60 71 73 3 14 25 36
 
Semi-magic III L9* 43 DOWN
49 60 71 73 3 14 25 36 38
597081 2 13 24 35 37 48
69801 12 23 34 45 47 58
79911 22 33 44 46 57 68
8 10 21 32 43 54 56 67 78
18 20 31 42 53 55 66 77 7
19 30 41 52 63 65 76 6 17
29 40 51 62 64 75 5 16 27
39 50 61 72 74 4 15 26 28

The Plane of Loubère Squares

At this point it may be said that alternatively these squares may be constructed using a plane of four squares. For example using the 7x7 square L7* 28 LEFT one can move up the right diagonal on a plane of four L7* 28 LEFT and generate the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.

This completes this section on regular and non-regular De La Loubère squares (Part I). The next section deals with a new Loubère-Knight method. To return to homepage.


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