New De La Loubère Method and Squares (Part I)
Regular and NonRegular Loubère Squares
The Full Monty I
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(n^{2} + 1). The properties of these regular or associated Loubère squares are:
 That the sum of the horizontal rows,
vertical columns and corner diagonals are equal to the magic sum S.
 The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to
n^{2} + 1, i.e., or twice the number in the center cell and are complementary to each other.
 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 
23  5  7 
14  16 
4  6  13 
20  22 
10  12  19 
21  3 
11  18  25 
2  9 

 
30  39  48 
1  10 
19  28 
38  47  7 
9  18 
27  29 
46  6  8 
17  26 
35  37 
5  14  16 
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 nonregular squares are produced. In fact, more nonregular squares are generated than
regular squares. This goes for this and other Loubère methods that I will be discussing. In addition, a new
LoubèreKnight 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
nonregular 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(n^{2}  n + 2) to ½(n^{2} + 7) for squares breaking
to the right or to the bottom.
 
1 
1  
 
 1  1 
 
 
1  1  
 
 
1   
 
 1 
  
 
1  1 
  
 1 
1  
  
1  1 
 


center Value
Equation  Value Left  Equation  Value Down 
½(n^{2}  5)  22  ½(n^{2} + 1)  25 
½(n^{2}  3)  23  ½(n^{2} + 3)  26 
½(n^{2}  1)  24  ½(n^{2} + 5)  27 
½(n^{2} + 1)  25  ½(n^{2} + 7)  28 
½(n^{2} + 3)  26  ½(n^{2}  5)  22 
½(n^{2} + 5)  27  ½(n^{2}  3)  23 
½(n^{2} + 7)  28  ½(n^{2}  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:
 Every number on the main diagonal is represented at least once in this type of square.
 Odd squares divisible by 3, i.e., 3(2n + 1) obey a simple modified rule.
These square may be magic or semimagic 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 nonregular Loubère squares
5x5 Squares
 To generate the nonregular 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.
 Move one cell to the left.
 Repeat the process until the nonregular square is filled, as shown below in squares 15.
 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.

⇒ 
2
6  5  
 
4   
11  10 
  
9  3 
  8 
2  
 7  1 
 

⇒ 
3
6  5  
 12 
4   
11  10 
 16  15 
9  3 
 14  8 
2  
13  7  1 
 

⇒ 
4
6  5  
18  12 
4   17 
11  10 
 16  15 
9  3 
20  14  8 
2  21 
13  7  1 
 19 

⇒ 
5L
6  5  24 
18  12 
4  23  17 
11  10 
22  16  15 
9  3 
20  14  8 
2  21 
13  7  1 
25  19 

≡ 
5D
19  21  3 
10  12 
25  2  9 
11  18 
1  8  15 
17  24 
7  14  16 
23  5 
13  20  22 
4  6 

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 nonregular Loubère square L7^{*} 28 LEFT from the
broken light blue diagonal set follows the same line.
1
 8  7 
 
 
14  6  
 
 15 
5   
 
 13 
  
 
12  4 
  
 11 
3  
  
10  2 
 
  9 
1  
 

⇒ 
2
16  8  7 
 
 24 
14  6  
 
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 
14  6  47 
39  31 
23  15 
5  46  38 
30  22 
21  13 
45  37  29 
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 

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 nonregular Semi Magic Loubère Squares
However, the use of odd squares divisible by three, as mention previously recycles through a triad of magic and semimagic
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 
57  68  79 
9  11 
22  33 
44  46 
67  78  8 
10  21 
32  43 
54  56 
77  7  18 
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 


Semimagic II L9^{*} 42 DOWN
48  59  70 
81  2 
13  24 
35  37 
58  69  80 
1  12 
23  34 
45  47 
68  79  9 
11  22 
33  44 
46  57 
78  8  10 
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 


Semimagic III L9^{*} 43 DOWN
49  60  71 
73  3 
14  25 
36  38 
59  70  81 
2  13 
24  35 
37  48 
69  80  1 
12  23 
34  45 
47  58 
79  9  11 
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 nonregular De La Loubère squares (Part I). The next section deals with a
new LoubèreKnight method. To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com