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

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
41	44	3	14	18	22	23
43	5	13	16	24	35	39
7	11	15	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
36	49	2	12	18	24	34
47	4	10	20	22	35	37
6	8	21	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
37	43	3	12	21	27	32
45	5	14	20	25	30	36
7	13	18	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
40	48	2	10	21	25	29
49	4	8	19	27	30	38
6	9	17	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
41	43	7	12	17	23	32
49	5	10	16	25	34	36
3	9	18	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
40	44	6	14	17	25	29
45	4	8	19	23	34	42
2	13	21	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
32	42	44	1	10	20	26
40	46	7	9	15	24	34
48	5	11	21	23	29	38
3	13	19	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
39	47	6	10	15	23	35
49	4	12	20	24	29	37
2	14	18	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	1	12	23	34	45
58	65	73	3	14	25	36	44	51
64	75	5	16	27	35	42	49	56
77	7	18	26	33	40	47	55	66
9	17	24	31	38	46	57	68	79
15	22	29	37	48	59	70	81	8
20	28	39	50	64	72	80	6	13
30	41	52	63	71	78	4	11	19
43	54	62	69	76	2	10	21	32

ID *Semi*-Magic
48	61	71	76	1	14	27	33	38
59	72	78	2	12	25	35	40	46
70	80	4	10	23	36	42	47	57
81	6	11	21	34	44	49	55	68
8	13	19	32	45	51	56	66	79
15	20	30	43	53	58	64	77	9
22	28	41	54	60	65	75	7	17
29	39	52	62	67	73	5	18	24
37	50	63	69	74	3	16	26	31

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	1	12	26	32	42
60	70	76	9	11	19	30	44	50
68	78	7	13	27	29	37	48	62
80	5	15	25	31	45	47	55	66
3	17	23	33	43	49	63	65	73
10	21	35	41	51	61	67	81	2
20	28	39	53	59	69	79	4	18
36	38	46	57	71	77	6	16	22
40	54	56	64	75	8	14	24	34

IIID *Semi*-Magic
51	59	71	75	1	11	27	31	43
61	69	77	8	12	19	29	45	49
67	79	6	14	26	30	37	47	63
81	4	16	24	32	44	48	55	65
2	18	22	34	42	50	62	66	73
10	20	36	40	52	60	68	80	3
21	28	38	54	58	70	78	5	17
35	39	46	56	72	76	7	15	23
41	53	57	64	74	9	13	25	33

9x9 Cell Values and S
Center Value of IC and ID	S + d	d	Center Value of IIIC and IIID	S + d	d
38	369	0	42	378	9
43	366	-3	41	369	0
40	372	3	44	396	27
41	369	0	39	381	-18
42	366	-3	37	333	-36
39	372	3	38	342	-27
44	369	0	45	405	36
37	366	-3	40	360	-9
45	372	3	43	387	18

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.

Center Value of IC and ID	S + d	d	Center Value of IIIC and IIID	S + d	d
38	369	0	42	378	9
43	366	-3	41	369	0
40	372	3	44	396	27
41	369	0	39	381	-18
42	366	-3	37	333	-36
39	372	3	38	342	-27
44	369	0	45	405	36
37	366	-3	40	360	-9
45	372	3	43	387	18

Center Value of IC and ID	S + d	d	Center Value of IIIC and IIID	S + d	d
38	369	0	42	378	9
43	366	-3	41	369	0
40	372	3	44	396	27
41	369	0	39	381	-18
42	366	-3	37	333	-36
39	372	3	38	342	-27
44	369	0	45	405	36
37	366	-3	40	360	-9
45	372	3	43	387	18

Center Value of IC and ID	S + d	d	Center Value of IIIC and IIID	S + d	d
38	369	0	42	378	9
43	366	-3	41	369	0
40	372	3	44	396	27
41	369	0	39	381	-18
42	366	-3	37	333	-36
39	372	3	38	342	-27
44	369	0	45	405	36
37	366	-3	40	360	-9
45	372	3	43	387	18