Method A-1 Variants 2, 3 and 4 for 7x7 Squares

A Discussion of Variant 2, 3 and 4

For the 7x7 examples, variant 2 is set up using the same method as variant 1 except that the left diagonal with the group of numbers ½ (n²-n+2) to ½(n²+n) in reverse order (top left corner to the right lower corner) from the numbers listed in the 7x7 complementary table. These numbers in the left diagonal correspond to 24 → 23 → 22 → 25 → 28 → 27 → 26. Variant 3 is set up in forward zigzag fashion and variant 4 in reverse zigzag fashion, the last two corresponding to 22 → 27 → 24 → 25 → 26 → 23 → 24 and 24 → 27 → 22 → 25 → 28 → 23 → 26, respectively. Alternatively, these may be shown as templates in the partial complementary tables where ⤩ and ⤧ point independently into two directions:

Variant 2
	↙	22	←	23	←	24
25
	↘	28	→	27	→	26

Variant 3
22		23		24	↘
	⤩		⤧			25
28		27		26	↙

Variant 4
	↙	22		23		24
25			⤧		⤩
	↘	28		27		26

The other diagonal, column and row of the wheel are then added using the templates obtained for the invert (variant 2), forward zigzag (variant 3) or invert zigzag (variant 4), followed by filling in of the "non-spoke" numbers. Below are the the results of filling up 7x7 squares for variant 2, 3 and 4 which follow the method of variant1.

The method uses parity to determine the pairs of squares to use as shown in the following three parity tables.
Also a symmetrical color scheme may be used to determine where to put the initial numbers of each of the pairs on the square and simplifies the method.
In these squares the legitimate colored cells to use are where light blue crosses pink or light blue crosses light blue as in variant 1.
It must be noted that entries into the non-spoke cells (obtained from the complementary tables) are in the following order for variants 2 and 3 (using as an example) 10 → 39 → 11 → 40 as opposed to variants 1 and 3 where the order is 10 → 40 → 11 → 39.


ROWS
1
2
3
5
6
7

Variant 2
SUM	Δ 175	PAIRS	PARITY
77	98	49+49	O+O
76	99	50+49	E+O
75	100	50+50	E+E
75	100	50+50	E+E
74	101	50+51	E+O
73	102	51+51	O+O

Variant 3
SUM	Δ 175	PAIRS	PARITY
75	100	50+50	E+E
74	101	50+51	E+O
77	98	49+49	O+O
73	102	51+51	O+O
76	99	49+50	E+O
75	100	50+50	E+E

Variant 4
SUM	Δ 175	PAIRS	PARITY
77	98	49+49	O+O
74	101	50+51	E+O
75	100	50+50	E+E
75	100	50+50	E+E
76	99	49+50	E+O
73	102	51+51	O+O

As an example, variant 2 is shown with the entries from the parity table included in the last two rows and columns and shows the crossover cells.

Variant 2

							175
24			7			46	77	98
	23		8		45		76	99
		22	9	44			75	100
49	48	47	25	3	2	1	175	0
		6	41	28			75	100
	5		42		27		74	101
4			43			26	73	102
77	76	75	175	75	74	73	175
98	99	100	0	100	101	102

⇒

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

⇒

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

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

Variant 3

22			7			46
	27		42		5
		24	9	44
49	2	47	25	3	48	1
		6	41	26
	45		8		23
4			43			28

⇒

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

⇒

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

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

Variant 4

24			9			44
	27		42		5
		22	7	46
47	2	49	25	1	48	3
		4	43	28
	45		8		23
6			41			26

⇒

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

⇒

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

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

The next page contains a 9x9 Variant .
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1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
																								25
49	48	47	46	45	44	43	42	41	40	39	38	37	36	35	34	33	32	31	30	29	28	27	26

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

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

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

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

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

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

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

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