New De La Loubère Pendulum Method (Part I)
Regular and NonRegular Squares
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 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 

Loubère squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time.
In this new method each subsequent number is added using the pendulum approach. When a break is encountered this may be a single move (right or down) or a knight move
as shown in the construction of the squares.
 Fill in the starting number and add the next numbers down the ladder followed by
up the ladder until the broken diagonal is filled tracing out an arc of a pendulum or
 Fill in the starting number and add the next numbers up the ladder followed by
down the ladder until the broken diagonal is filled again tracing out a pendular arc.
In the first group of new Loubère squares, which I will label PLn^{*} (center cell#) [LD or RU,small or large arc]
where PLn^{*} signifies a small or large arc pendulum move nxn, left down or up right,
Loubère square with a certain center cell number, breaking either down or to
the left depending where on the diagonal the opening 1 resides on.
In the second group
the squares are labeled PLKn^{*} (center cell#) [LD,small or large arc,(n_{1},n_{2})]
where PLKn^{*} signifies a small or large arc pendulum move nxn Loubère square with a certain center cell number,
breaking in knight fashion. n_{1} or n_{2} may be either
a = ½(n 3) or b = ½(n 1) and
n is the order of the square. This method will be shown starting on this page and continuing in
new Loubère Knightbreak pendulum squares method (Part II).
Construction of the 5x5 Loubère Pendulum Magic Squares
5x5 Small Arc Squares
Group IA
 To generate the regular square, PL5^{*} 13 [DL, small arc],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first down left, then up right continuously switching
until blocked by a previous number.
 Move one cell down.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

⇒ 
3
  1 
7  
 2  9 
 
4  10  
 
8  11  
 5 
12   
3  6 

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

⇒ 
5 PL5^{*} 13 [DL,small arc]
20  23  1 
7  14 
21  2  9 
15  18 
4  10  13 
16  22 
8  11  17 
24  5 
12  14  25 
3  6 

Group IB
 To generate the nonregular magic square, PL5^{*} 14 [RU, small arc],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first up right, then down left continuously switching
until blocked by a previous number.
 Move one cell right.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

⇒ 
3
  1 
9  
 3  7 
 
5  6  
 
8  12  
 4 
11   
2  10 

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

⇒ 
5 PL5^{*} 14 [RU,small arc]
17  25  1 
9  13 
24  3  7 
15  16 
5  6  14 
18  22 
8  12  20 
21  4 
11  19  23 
2  10 

5x5 Big Arc Squares
Group IIA
 To generate the nonregular magic square, PL5^{*} 12 [RU, big arc],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first up right, then down left continuously switching
until blocked by a previous number.
 Move one cell down.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

⇒ 
3
  1 
7  
 5  9 
 
3  6  12 
 
10   
 2 
11   
4  8 

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

⇒ 
5 PL5^{*} 12 [RU,big arc]
19  23  1 
7  15 
22  5  9 
13  16 
3  6  12 
20  24 
10  14  18 
21  2 
11  17  25 
4  8 

Group IIB
 To generate the nonregular magic square, PL5^{*} 15 [DL, big arc],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first down left, then up right continuously switching
until blocked by a previous number.
 Move one cell right.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

⇒ 
3
  1 
9  12 
 4  7 
 
2  8  
 
10  11  
 3 
  
5  6 

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

⇒ 
5 PL5^{*} 15 [DL,big arc]
18  25  1 
9  12 
21  4  7 
13  20 
2  8  15 
16  24 
10  11  19 
22  3 
14  17  23 
5  6 

7x7 Loubère Pendulum Squares Groups IAIIB
Fully constructed 7x7 Loubère pendulum squares one each and typical one move break are shown below. Again these squares belong to the set situated on the
yellow diagonal.
IA
30  41  47 
1  11 
17  28 
42  45  2 
13  19 
22  32 
43  4  14 
17  23 
34  40 
6  12  15 
25  35 
38  44 
10  16  27 
33  36 
46  7 
18  28  31 
37  48 
5  8 
26  29  39 
49  3 
9  20 


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


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


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

Construction of 5x5 KnightBreak Loubère Small and Large Arc Pendulum Magic Squares
These new Loubère squares, which I will label PLKn^{*} (center cell#) [LD or RU,small or
large arc,(n_{1}U,n_{2}R)] where PLKn^{*} signifies a small or large arc pendulum move. nxn
is a Loubère square with a certain cell number, breaking in knight fashion down or to
the left depending where on the diagonal the opening 1 resides on.
n_{1} or n_{2} are the variable numbers a or b shown below.
This set of squares consists of six groups of small and large arc pendulum squares according to the following table:
where a = ½(n 3) and b = ½(n 1) and n
is the order of the square.
Group C  Group D 
I (small arc)  (bD,aL)  I (small arc) 
(aU,bR) 
II (large arc)  (aU,bR)  II (large arc) 
(bD,aL) 
III (large arc)  (bD,aL)  III (large arc) 
(aU,bR) 
Group IC
 To generate the nonregular magic square, PLK5^{*} 12
[RU,small arc,(2D,1L)],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first up right, then down left continuously switching
until blocked by a previous number.
 Move in knight fashion two cells down and one cell left.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

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

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

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

Group ID
 To generate the nonregular magic square, PLK5^{*} 15
[LD,small arc,(1U,2R)],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first down left, then up right continuously switching
until blocked by a previous number.
 Move in knight fashion one cell up and two cells right.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

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

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

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

Group IIC
No magic square.
Group IID
No magic square.
Group IIIC
 To generate the nonregular magic square, PLK5^{*} 15
[RU,large arc,(2D,1L)],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first down left, then up right continuously switching
until blocked by a previous number.
 Move in knight fashion one cell up and two cells right.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

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

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

⇒ 
5 SemiMagic
20  22  1 
8  14 
24  5  7 
11  18 
3  9  15 
17  21 
6  13  19 
55  2 
12  16  23 
4  10 

Group IIID
 To generate the nonregular magic square, PLK5^{*} 14
[LD,large arc,(1U,2L)],
place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first down left, then up right continuously switching
until blocked by a previous number.
 Move in knight fashion one cell up and two cells right.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

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

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

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

This completes this section on De La Loubère pendulum squares (Part I). The next section deals with a continuation of these
new Loubère Knightbreak pendulum squares method (Part II). To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com