NEW MAGIC SQUARES WHEEL METHOD

Part VII

9x9 Magic Square Wheel

A magic square is an arrangement of numbers 1,2,3,... n² where every row, column and diagonal add up to the same magic sum S and n is also the order of the square. A magic square having all pairs of cells diametrically equidistant from the center of the square and equal to the sum of the first and last terms of the series n² + 1 is also called associated or symmetric. In addition, the center of this type of square must always contain the middle number of the series, i.e., ½(n² + 1).

A second modified facile method for the construction of wheel type magic squares is now available. The position of the spokes are rotated by 90° so that the left diagonal starts at the bottom left cell. The 5x5 square is first filled followed by the 7x7 and finally the 9x9. The 9x9 square as well as the 3x3, 5x5 and 7x7 squares are magic and thus classified as border. In addition the bordered square may be everted to give an opposite square which is no longer bordered.

The new magic squares with n = 9 are constructed as follows using a complimentary table as a guide.

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

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42

A 9x9 Transposed Magic Square Using the Diagonals {37,38,39,40,41,42,43,44,45} and {2,5,8,11,41,71,74,77,80}

The 9x9 square is to be filled with 33 numbers from the subset 1-12 and their complements 70-81 and the numbers 37-45. The spokes of the wheel are generated as follows: Numbers 37-45 in the left diagonal; numbers 2,5,8,11 and conjugates 80,77,74,71 in the right diagonal; numbers 1,4,7,10 and conjugates 81,78,75,72 in top to bottom center; and 3,6,9,12 and conjugates 70,73,76,79 in center horizontal (square A1). The addition of these pair of numbers and conjugates to the 9x9 square are shown below using directional pointed arrows:

1 4 7 10 2 8 6 11 3 6 9 12 ... 37 38 39 40

41

81 78 75 72 80 77 74 71 79 76 73 70 ... 45 44 43 42

↓ ↖ → ... ↗
Sum up the rows and columns 1-4 and 6-9 and subtract from the magic sum 369. This gives the amounts required (shown in green Square A2). The last column shows the two amounts need to complete the row and column (shown in yellow).
Using adjacent pair numbers from the complementary table above, fill in the non-spoke cells of the 5x5 square, then the 7x7 and finally the 9x9 using the inset below as a guide: (Square A3, A4 and A5).

A6 shows the square in border form.

A1
80				1				45
	77			4			44
		74		7		43
			71	10	42
3	6	9	12	41	70	73	76	79
			40	72	11
		39		75		8
	38			78			5
37				81				2

⇒

A2
80				1				45	243	81x3
	77			4			44		244	82+81x2
		74		7		43			245	82x2+81
			71	10	42				246	82x2+83
3	6	9	12	41	70	73	76	79
			40	72	11				246	82x3
		38		80		6			247	82x2+83
	39			75			8		248	83x2+82
37				81				2	249	83x3
249	248	247	246		246	245	244	243

⇒

A3
80				1				45
	77			4			44
		74	13	7	68	43
		67	71	10	42	15
3	6	9	12	41	70	73	76	79
		16	40	72	11	66
		39	69	75	14	8
	38			78			5
37				81				2

⇒

A4
80				1				45
	77	17	19	4	62	64	44
	61	74	13	7	68	43	21
	59	67	71	10	42	15	23
3	6	9	12	41	70	73	76	79
	24	16	40	72	11	66	58
	22	39	69	75	14	8	60
	38	65	63	78	20	18	5
37				81				2

⇒

A5
80	25	27	29	1	52	54	56	45
51	77	17	19	4	62	64	44	31
49	61	74	13	7	68	43	21	33
47	59	67	71	10	42	15	23	35
3	6	9	12	41	70	73	76	79
36	24	16	40	72	11	66	58	46
34	22	39	69	75	14	8	60	48
32	38	65	63	78	20	18	5	50
37	57	55	53	81	30	28	26	2

⇒

A6 Border
80	25	27	29	1	52	54	56	45
51	77	17	19	4	62	64	44	31
49	61	74	13	7	68	43	21	33
47	59	67	71	10	42	15	23	35
3	6	9	12	41	70	73	76	79
36	24	16	40	72	11	66	58	46
34	22	39	69	75	14	8	60	48
32	38	65	63	78	20	18	5	50
37	57	55	53	81	30	28	26	2

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	26

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62	61	60	59	58	57	56

						27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
						55	54	53	52	51	50	49	48	47	46	45	44	43	42

Conversion of the 9x9 into its transposed opposite

Using the method of Part IV, the 9x9 transposed opposite generates a new square which is not a border square. Only the external square is magic.

This completes Part VII of a 9x9 border Magic Square Wheel method.
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1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42

1	4	7	10	2	8	6	11	3	6	9	12	...	37	38	39	40
																	41
81	78	75	72	80	77	74	71	79	76	73	70	...	45	44	43	42


↓				↖				→				...	↗

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	26

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62	61	60	59	58	57	56

						27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
						55	54	53	52	51	50	49	48	47	46	45	44	43	42

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

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42

1	4	7	10	2	8	6	11	3	6	9	12	...	37	38	39	40
																	41
81	78	75	72	80	77	74	71	79	76	73	70	...	45	44	43	42


↓				↖				→				...	↗

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	26

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62	61	60	59	58	57	56

						27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
						55	54	53	52	51	50	49	48	47	46	45	44	43	42

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

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42

1	4	7	10	2	8	6	11	3	6	9	12	...	37	38	39	40
																	41
81	78	75	72	80	77	74	71	79	76	73	70	...	45	44	43	42


↓				↖				→				...	↗

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	26

81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62	61	60	59	58	57	56

						27	28	29	30	31	32	33	34	35	36	37	38	39	40
																				41
						55	54	53	52	51	50	49	48	47	46	45	44	43	42