Continuation of New Méziriac Knight-Break Pendulum Method (Part IV)

These new Méziriac squares, which I will label PMn^* (center cell#) [LD or RU,small or large arc,(n₁U,n₂R)] where PMn^* signifies a small or large arc pendulum move. nxn is a Méziriac 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)	(aU,bL)	I (small arc)	(bU,aL)
II (large arc)	(bU,aL)	II (large arc)	(aU,bL)
III (large arc)	(aU,bL)	III (large arc)	(bU,aL)

Construction of 5x5 Knight-Break Méziriac Small Arc Pendulum Magic Squares

Group IC

To generate the semi-magic square, PMK5^* 14 [RU,small 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 up right, then down left continuously switching until blocked by a previous number.
Move in knight fashion one cell up and two cells left.
Repeat the process until the square is filled, as shown below in squares 1-5.

1
4
				2
			1
		3		6
	5

⇒

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

⇒

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

⇒

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

⇒

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

********************************************************************************************************************************************************

Group ID

To generate the semi-magic square, PMK5^* 15 [LD,small arc,(2U,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 two cells up and one cell left.
Repeat the process until the square is filled, as shown below in squares 1-5.

1
5
				3
			1
		2		6
	4

⇒

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

⇒

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

⇒

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

⇒

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

********************************************************************************************************************************************************

Construction of 5x5 Knight-Break Méziriac Large Arc Pendulum Magic Squares

Group IIC

To generate the magic square, PMK5^* 13 [RU,large arc,(2U,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 two cells up and one cell left.
Repeat the process until the square is filled, as shown below in squares 1-5.

1
2
	6			4
			1
		5
	3

⇒

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

⇒

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

⇒

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

⇒

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

********************************************************************************************************************************************************

Group IID

To generate the magic square, PMK5^* 15 [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 left.
Repeat the process until the square is filled, as shown below in squares 1-5.

1
3		6
				5
			1
		4
	2

⇒

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

⇒

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

⇒

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

⇒

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

********************************************************************************************************************************************************

Group IIIC

PMK5^* 11 [RU,large arc,(1U,2L)], No magic squares possible.

Group IIID

PMK5^* 14 [LD,large arc,(2U,1L)], No magic squares possible.

********************************************************************************************************************************************************

7x7 Méziriac Knight Pendulum Squares Groups IC-IID

Fully constructed 7x7 knight 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.

Group IC and Group ID

PMK7^* 27 [RU,small arc,(2U,3L)]

PMK7^* 28 [LD,small arc,(3U,2L)]

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

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

********************************************************************************************************************************************************

Group IIC and Group IID

PMK7^* 27 [RU,large arc,(3U,2L)]

PMK7^* 26 [LD,large arc,(2U,3L)]

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

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

********************************************************************************************************************************************************

Group IIIC and Group IIID

PMK7^* 24 [RU,large arc,(2U,3L)]

PMK7^* 25 [LD,large arc,(3U,2L)]

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

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

********************************************************************************************************************************************************

9x9 Méziriac Knight Pendulum Squares Groups IC-IIID

Fully constructed 9x9 knight 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. These three groups are semi-magic and only groups I contain at least one magic square with S of 369 as shown in the last table.

Group IC, Group ID, Group IIID

PMK9^* 44 [RU,small arc,(3U,4L)]

PMK9^* 45 [LD,small arc,(4U,3L)]

PMK9^* 42 [LD,large arc,(4U,3L)]

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

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

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

9x9 Cell Values and S
Center Value of IC or ID	S + d	d	Center Value of IIID	S + d	d
45	405	36	42	378	9
37	333	-36	41	354	-15
44	396	27	44	375	6
39	351	-18	39	378	9
42	378	9	37	354	-15
41	369	0	38	375	6
40	360	-9	45	378	9
43	387	18	40	354	-15
38	342	-27	43	375	6

This completes this section on Méziriac knight-break pendulum squares (Part IV). To see a new Méziriac regular and variable knight-break full pendulum methods (Part II). To return to homepage.

Center Value of IC or ID	S + d	d	Center Value of IIID	S + d	d
45	405	36	42	378	9
37	333	-36	41	354	-15
44	396	27	44	375	6
39	351	-18	39	378	9
42	378	9	37	354	-15
41	369	0	38	375	6
40	360	-9	45	378	9
43	387	18	40	354	-15
38	342	-27	43	375	6

Center Value of IC or ID	S + d	d	Center Value of IIID	S + d	d
45	405	36	42	378	9
37	333	-36	41	354	-15
44	396	27	44	375	6
39	351	-18	39	378	9
42	378	9	37	354	-15
41	369	0	38	375	6
40	360	-9	45	378	9
43	387	18	40	354	-15
38	342	-27	43	375	6

Center Value of IC or ID	S + d	d	Center Value of IIID	S + d	d
45	405	36	42	378	9
37	333	-36	41	354	-15
44	396	27	44	375	6
39	351	-18	39	378	9
42	378	9	37	354	-15
41	369	0	38	375	6
40	360	-9	45	378	9
43	387	18	40	354	-15
38	342	-27	43	375	6