New Loubère Full Pendulum Method (Part I)

A pendulum

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.

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 full 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 1 and add the next numbers consecutively down the ladder followed by consecutively up the ladder until the broken diagonal is filled tracing out a full arc of a pendulum or
  2. Fill in the starting number 1 and add the next numbers consecutively up the ladder followed by consecutively down the ladder until the broken diagonal is filled again tracing out a full arc of a pendulum.

In the first group of new Loubère squares, which I will label PLn* (center cell#) [LD or RU,full arc] where PLn* signifies a full arc pendulum move nxn, left down or up right, Loubère square with the center cell number of the square and breaking either down or to the right. In addition, all odd squares, except those divisible by three, are magic.

In the second group the squares are labeled PLKn* (center cell#) [LD or RU,full arc,(n1,n2)] where PLKn* signifies a full arc pendulum move nxn Loubère square with a center cell number and breaking in knight fashion. (n1, n2) may be either (1 up, 2 right) or (2 down, 1 left). In addition, all odd squares, except those divisible by three, are magic.

Construction of the 5x5 Loubère Full Pendulum Magic Squares

5x5 Full Arc Squares

Group IA
  1. To generate the square, PL5* 14 [LD, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively first down left filling the partial diagonal, then up right consecutively filling the rest of the diagonal 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
5
4 6
2
1 7
28
310
911 5
4 6
3
1 7 13
28 15
31014 16
911 5
12 4 6
4
20 1 7 13
2128 15 19
31014 16
91117 5
12 18 4 6
5 PL5* 14 [LD,full arc]
20 24 1 7 13
2128 15 19
31014 16 22
91117 23 5
12 18 25 4 6
Group IB
  1. To generate the magic square, PL5* 13 [RU, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively first up then right, filling up the broken diagonal 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
4
56
3
2
2
1 8
47
56
9 3
11 2 10
3
1 8 14
47 15 16
5613
912 3
11 2 10
4
17 1 8 14
47 15 16
5613 19
91220 21 3
11 18 2 10
5 PL5* 13 [RU,full arc]
17 25 1 8 14
2347 15 16
5613 19 22
91220 21 3
11 18 24 2 10

7x7 Full Arc Squares

Group IA
  1. To generate the magic square, PL7* 24 [LD, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively down left, then consecutively down left 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 1-4.
1
110
2 11
314
4 13 15
12 7
68
5 9
2
110 21 26
2 11 20 22
314 19 23
4 13 15 24
121625 7
17 28 68
27 29 5 918
3
30 39 110 21 26
422 11 20 2231
43314 19 23 3241
4 13 15 24 35 40
121625 34 36 7
17 28 33 37 68
27 29 38 5 918
4 PL7* 24 [LD,full arc]
30 39 48 110 21 26
42472 11 20 2231
43314 19 23 3241
4 13 15 24 35 4044
121625 34 36 457
17 28 33 37 46 68
27 29 38 49 5 918
Group IB
  1. To generate the magic square, PL7* 26 [RU, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively up right, then consecutively down left 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 1-4.
1
111
5 10
69
7 8
12 4
313
2 1415
2
111 19 24
5 10 20 23
69 21 22
7 8 18 26
121727 4
16 28 29 313
25 2 1415
3
34 37 111 19 24
365 10 20 2335
69 21 22 3240
7 8 18 26 31 41
121727 30 42 434
16 28 29 39 313
25 33 38 2 1415
4 PL5* 26 [RU,full arc]
34 37 49 111 19 24
36465 10 20 2335
4569 21 22 3240
7 8 18 26 31 4144
121727 30 42 434
16 28 29 39 47 313
25 33 38 48 2 1415

Construction of 5x5 Knight-Break Loubère Full Arc Pendulum Magic Squares

These new Loubère squares, which I will label PLKn* (center cell#) [LD or RU,small or large arc,(U,R or D,L)] where PLKn* signifies a small or large arc pendulum move. nxn is a Loubère square with a certain cell number, breaking in knight fashion (up,right) or (down,left)

Group IC
  1. To generate the magic square, PLK5* 15 [LD,full 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 consecutively down left, then consecutively up right 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.
  4. Note that PLK5* 15 [LD,full arc,(2D,1L)] does not generate a magic square.
1
1
2
36
5
4
2
1 10
29
36
7 5
11 4 8
3
1 10 12
29 13 16
3615
714 5
11 4 8
4
19 1 10 12
29 13 16
3615 17
71418 21 5
11 20 4 8
5
19 23 1 10 12
2529 13 16
3615 17 24
71418 21 5
11 20 22 4 8
Group ID
  1. To generate the magic square, PLK5* 12 [RU,full 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 consecutively up right, then consecutively left down 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
4
5
3
2 6
2
1 9
410
58
711 3
2 6
3
1 9 15
410 13
5812 16
711 3
14 2 6
4
18 1 9 15
21410 13 17
5812 16
71119 3
14 20 2 6
5
18 22 1 9 15
21410 13 17
5812 16 24
71119 25 3
14 20 23 2 6
Group IC
  1. To generate the magic square, PLK7* 23 [LD,full 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 consecutively down left, then consecutively up right 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.
  4. Note that PLK7* 26 [LD,full arc,(2D,1L)] in this case generates a magic square.
1
114
2 13
312
4 8
9 7
610
5 1115
2
114 16 27
2 13 17 26
312 18 22
4 8 21 23
92024 7
19 25 29 610
28 5 1115
3
31 40 114 16 27
362 13 17 2632
312 18 22 3537
4 8 21 23 34 38
92024 33 39 437
19 25 29 42 610
28 30 41 5 1115
4
31 40 40 114 16 27
36492 13 17 2632
48312 18 22 3537
4 8 21 23 34 3847
92024 33 39 437
19 25 29 42 44 610
28 30 41 45 5 1115
Group ID
  1. To generate the magic square, PLK7* 27 [RU,full arc,(2D,1L)], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing continuously diagonally up right, then continuously down left 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 1-4.
1
113
5 14
611
7 10 15
9 4
38
2 12
2
113 18 23
5 14 17 22
611 16 26
7 10 15 27
91928 4
20 25 38
24 29 2 1221
3
33 42 113 18 23
395 14 17 2234
43611 16 26 3538
7 10 15 27 32 37
91928 31 36 4
20 25 30 40 38
24 29 41 2 1221
4
33 42 45 113 18 23
39445 14 17 2234
43611 16 26 3538
7 10 15 27 32 3747
91928 31 36 484
20 25 30 40 49 38
24 29 41 46 2 1221

This completes this section on De La Loubère full pendulum squares (Part I). The next section deals with new Méziriac full pendulum squares method (Part II). To return to homepage.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com