New De La Loubère Pendulum Method (Part I)

Regular and Non-Regular Squares

A pendulum

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.

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

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.

  1. 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
  2. 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,(n1,n2)] where PLKn* signifies a small or large arc pendulum move nxn Loubère square with a certain center cell number, breaking in knight fashion. n1 or n2 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 Knight-break pendulum squares method (Part II).

Construction of the 5x5 Loubère Pendulum Magic Squares

5x5 Small Arc Squares

Group IA
  1. 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.
  2. Move one cell down.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
2
 
 
  3
2
1 7
2
4
8 5
3 6
3
1 7
29
410
811 5
12 3 6
4
20 1 7 14
2129 15 18
41013 16
81117 5
12 14 3 6
5 PL5* 13 [DL,small arc]
20 23 1 7 14
2129 15 18
41013 16 22
81117 24 5
12 14 25 3 6
Group IB
  1. To generate the non-regular 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.
  2. Move one cell right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
3
 
 
2
2
1
37
56
8 4
2
3
1 9
37
56
812 4
11 2 10
4
17 1 9 13
37 15 16
5614 18
81220 21 4
11 2 10
5 PL5* 14 [RU,small arc]
17 25 1 9 13
2437 15 16
5614 18 22
81220 21 4
11 19 23 2 10

5x5 Big Arc Squares

Group IIA
  1. To generate the non-regular 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.
  2. Move one cell down.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
 
3
2
 
2
1 7
5
36
2
4 8
3
1 7
59
3612
10 2
11 4 8
4
19 1 7 15
59 13 16
3612 20
101418 21 2
11 17 4 8
5 PL5* 12 [RU,big arc]
19 23 1 7 15
2259 13 16
3612 20 24
101418 21 2
11 17 25 4 8
Group IIB
  1. To generate the non-regular 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.
  2. Move one cell right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
 
2
3
 
2
1
47
28
3
5 6
3
1 9 12
47
28
1011 3
5 6
4
18 1 9 12
2147 13 20
2815 16
101119 3
14 17 5 6
5 PL5* 15 [DL,big arc]
18 25 1 9 12
2147 13 20
2815 16 24
101119 22 3
14 17 23 5 6

7x7 Loubère Pendulum Squares Groups IA-IIB

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
42452 13 19 22 32
43414 17 23 34 40
61215 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
36483 11 19 23 35
4659 21 22 34 38
7820 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
39457 13 16 26 29
4458 18 24 35 41
31420 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
42436 11 16 24 33
4849 17 26 35 36
21019 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 Knight-Break 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,(n1U,n2R)] 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. n1 or n2 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
  1. To generate the non-regular 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.
  2. Move in knight fashion two cells down and one cell left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
3
5
4
2 6
2
1 8
310
59
711 4
2 6
3
1 8 15
310 14
5912 16
711 4
13 2 6
4
19 1 8 15
21310 14 17
5912 16
71118 4
13 20 2 6
5
19 22 1 8 15
21310 14 17
5912 16 23
71118 25 4
13 20 24 2 6
Group ID
  1. To generate the non-regular 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.
  2. Move in knight fashion one cell up and two cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
2
46
5
3
2
1 10
28
46
7 5
11 3 9
3
1 10 12
28 14 16
4615
713 5
11 3 9
4
18 1 10 12
28 14 16
4615 17
71319 21 5
11 20 3 9
5
18 24 1 10 12
2528 14 16
4615 17 23
71319 21 5
11 20 22 3 9
Group IIC

No magic square.

Group IID

No magic square.

Group IIIC
  1. To generate the non-regular 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.
  2. Move in knight fashion one cell up and two cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
5
3
6 2
4
2
1 8
57 11
39
6 2
4 10
3
1 8 14
57 11
3915
613 2
12 16 4 10
4
20 1 8 14
57 11 18
3915 17 21
61319 2
12 16 4 10
5 Semi-Magic
20 22 1 8 14
2457 11 18
3915 17 21
61319 55 2
12 16 23 4 10
Group IIID
  1. To generate the non-regular 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.
  2. Move in knight fashion one cell up and two cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
4
2
6 3
5
2
1 7
48 11
210
6 3
5 9
3
1 7 15
48 11
21014
612 3
13 16 5 9
4
19 1 7 15
48 11 17
21014 18 21
61220 3
13 16 5 9
5 Semi-Magic
19 23 1 7 15
2548 11 17
21014 18 21
61220 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 Knight-break pendulum squares method (Part II). To return to homepage.


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