Continuation of New De La Loubère Variable Knight-Break Pendulum Method (Part II)

A pendulum

A Discussion of the New Methods

7x7 Knight Loubère Pendulum Squares Groups IC-ID

Before proceeding into variable knight moves it is also possible to generate magic squares using only (1,2) or (2,1) knight moves. For a fully constructed 7x7 knight Loubère pendulum squares a typical regular knight break move generates only the two squares below. Again these squares belong to the set situated on the yellow diagonal.

IC
31 42 46 1 12 20 23
41443 14 18 22 23
43513 16 24 35 39
71115 26 34 37 45
9 17 28 32 36 47 6
19 27 30 38 49 4 8
25 29 40 48 2 10 21
 
ID
32 38 48 1 14 16 26
36492 12 18 24 34
47410 20 22 35 37
6821 23 33 39 45
9 19 25 31 41 43 7
17 27 29 42 44 5 11
28 30 40 46 3 13 15

As previously mentioned, in Part I these new Loubère squares, which I labeled PLKn* (center cell#) [LD or RU,small or large arc,(n1U,n2R)] where PMn* 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)

7x7 Loubère Knight Pendulum Squares Groups IC-IID

Fully constructed 7x7 knight-break Loubère pendulum squares one each and typical one move break are shown below. The number  2  is shown in green so as to distinguish which way the pendulum is initially moving. In addition the variable letters a and b may be 2 or 3.

Group IC and Group ID
PLK7* 23 [RU,small arc,(3D,2L)]   PLK7* 28 [LD,small arc,(2U,3R)]
IC Magic
34 39 44 1 10 19 28
37433 12 21 27 32
45514 20 25 30 36
71318 23 29 38 47
11 16 22 31 40 49 6
15 24 33 42 48 4 9
26 35 41 46 2 8 17
 
ID Magic
31 43 46 1 12 20 23
40482 10 21 25 29
4948 19 27 30 38
6917 28 32 36 47
11 15 26 34 37 45 7
16 24 35 39 43 5 13
22 33 41 44 3 14 18
Group IIC and Group IID
PLK7* 27 [RU,large arc,(2U,3R)]   PLK7* 24 [LD,large arc,(3D,2L)]
IIC Magic
30 39 48 1 14 19 24
41437 12 17 23 32
49510 16 25 34 36
3918 27 29 42 47
11 20 22 35 40 45 2
15 28 33 38 44 4 13
26 31 37 46 6 8 21
 
IID Magic
35 38 46 1 12 16 27
40446 14 17 25 29
4548 19 23 34 42
21321 24 32 36 47
11 15 26 30 41 49 3
20 28 31 39 43 5 9
22 33 37 48 7 10 18
Group IIIC and Group IIID
PLK7* 25 [RU,large arc,(3D,2L)]   PLK7* 26 [LD,large arc,(2U,3R)]
IIIC Semi-Magic
324244 1 10 20 26
40467 9 15 24 34
48511 21 23 29 38
31319 25 35 37 43
8 17 27 33 39 49 2
16 22 31 41 47 4 14
28 30 36 45 6 12 18
 
IIID Semi-Magic
33 41 45 1 9 21 25
39476 10 15 23 35
49412 20 24 29 37
21418 26 34 38 43
8 16 28 32 40 48 3
17 22 30 42 46 5 13
27 31 36 44 7 11 19

9x9 Loubère Variable Knight-Break Pendulum Squares Groups IC-IIID

Fully constructed 9x9 knight-break Loubère pendulum squares and typical knight break moves are shown below the green  2  shows the direction of swing of the pendulum. Group II squares are non-magic. Again these squares belong to the set situated on the yellow diagonal.

Group IC, Group ID
PLK9* 44 [RU,small arc,(4D,3L)]   PLK9* 45 [LD,small arc,(3U,4R)]
IC Semi-Magic
53 60 67 74 112 233445
58 65 73 3 1425 364451
64 75 5 16 2735 424956
77 7 18 26 3340 475566
91724 3138 46 57 6879
15 22 29 37 4859 70818
20 28 39 50 6472 80613
30 41 52 63 7178 41119
43 54 62 69 762 102132
 
ID Semi-Magic
48 61 71 76 114 273338
59 72 78 2 1225 354046
70 80 4 10 2336 424757
81 6 11 21 3444 495568
81319 3245 51 56 6679
15 20 30 43 5358 64779
22 28 41 54 6065 75717
29 39 52 62 6773 51824
37 50 63 69 743 162631
Group IIIC, Group IIID
PLK9* 43 [RU,small arc,(4D,3L)]        PLK9* 42 [LD,small arc,(3U,4R)]
IIIC Semi-Magic
52 58 72 74 112 263242
60 70 76 9 1119 304450
68 78 7 13 2729 374862
80 5 15 25 3145 475566
31723 3343 49 63 6573
10 21 35 41 5161 67812
20 28 39 53 5969 79418
36 38 46 57 7177 61622
40 54 56 64 758 142434
 
IIID Semi-Magic
51 59 71 75 111 273143
61 69 77 8 1219 294549
67 79 6 14 2630 374763
81 4 16 24 3244 485565
21822 3442 50 62 6673
10 20 36 40 5260 68803
21 28 38 54 5870 78517
35 39 46 56 7276 71523
41 53 57 64 749 132533
9x9 Cell Values and S
Center Value of IC and IDS + ddCenter Value of IIIC and IIIDS + dd
383690423789
43366-3413690
4037234439627
41369039381-18
42366-337333-36
39372338342-27
4436904540536
37366-340360-9
4537234338718

This completes this section on De La Loubère pendulum squares (Part I). The next section deals with a new Méziriac pendulum squares method (Part III). To see a new Loubère regular and knight-break full pendulum methods (Part I). To return to homepage.


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