Octagon Algorithm for Wheel Magic Squares (Part III)

The Eight Node Way - 9×9 Squares

The previous section Part II introduced the Octagon A algorithm for the construction of 7×7 Wheel Squares. This section will cover the 9th order squares. The square will be built according to Part I starting with the initial node at top left of the octagon.

The squares will be constructed, however, with only partial coding for the wheel portion but not for the complete magic square since the octagon method requires an inordinate amount of code. The reason being that while and for loops cannot be used to fill in the cells as discussed in Part I.

Six 9×9 examples are shown below. Starting with the construction of the Wheel 9(0)-1 one may use the Octagon A algorithm to fill in the rest of the white cells a number of ways. Magic Square 9(0)-1 is generated by initially placing the number 13 in the second top cell [position (0,1)] then adding the rest of the numbers as shown. Note that position is defined here as (row,column) where the first cell is (0,0) and the grey cells are sums, where every row, column and diagonals must add up at the end to the magic sum 369.

Wheel 9(0)-1
77				1				45	123
	76			2			44		122
		75		3		43			121
			74	4	42				120
9	10	11	12	41	70	71	72	73	369
			40	78	8				122
		39		79		7			121
	38			80			6		124
37				81				5	123
123	124	125	126	369	120	121	122	123	369

⇒

Magic Square 9(0)-1
77	13	17	21	1	61	65	69	45	369
16	76	25	29	2	53	57	44	67	369
20	28	75	33	3	49	43	55	63	369
24	32	36	74	4	42	47	51	59	369
9	10	11	12	41	70	71	72	73	369
58	50	46	40	78	8	35	31	23	369
62	54	39	48	79	34	7	27	19	369
66	38	56	52	80	30	26	6	15	369
37	68	64	60	81	22	18	14	5	369
369	369	369	369	369	369	369	369	369	369

Magic Square 9(0)-2 is generated by initially placing the number 13 in the fourth top cell [position (0,3)] then working backwards to fill in the cells. Magic Square 9(0)-3 is generated by placing the 13 internally in a green cell [position (2,3)] and working outward towards the perimeter of the square.

Magic Square 9(0)-2
77	21	17	13	1	69	65	61	45	369
24	76	29	25	2	57	53	44	59	369
20	32	75	33	3	49	43	51	63	369
16	28	36	74	4	42	47	55	67	369
9	10	11	12	41	70	71	72	73	369
66	54	46	40	78	8	35	27	15	369
62	50	39	48	79	34	7	31	19	369
58	38	52	56	80	26	30	6	23	369
37	60	64	68	81	14	18	22	5	369
369	369	369	369	369	369	369	369	369	369

Magic Square 9(0)-3
77	25	29	33	1	49	53	57	45	369
28	76	17	21	2	61	65	44	55	369
32	20	75	13	3	69	43	63	51	369
36	24	16	74	4	42	67	59	47	369
9	10	11	12	41	70	71	72	73	369
46	58	66	40	78	8	15	23	35	369
50	62	39	68	79	14	7	19	31	369
54	38	64	60	80	22	18	6	27	369
37	56	52	48	81	34	30	26	5	369
369	369	369	369	369	369	369	369	369	369

Magic Square 9(1)-1 and 9(2)-1 are generated using a different wheel for each square (computer code below) and then filling in the square starting at position (0,1) with the number 13 as was done with Magic Square 9(0)-1.

Magic Square 9(1)-1
75	1	17	21	3	61	65	81	45	369
16	74	25	29	4	53	57	44	67	369
20	28	73	33	5	49	43	55	63	369
24	32	36	72	6	42	47	51	59	369
11	12	13	14	41	68	69	70	71	369
58	50	46	40	76	10	35	31	23	369
62	54	39	48	77	34	9	27	19	369
66	38	56	52	78	30	26	8	15	369
37	80	64	60	79	22	18	2	7	369
369	369	369	369	369	369	369	369	369	369

Magic Square 9(2)-1
73	1	17	21	5	61	65	81	45	369
4	72	25	29	6	53	57	44	79	369
20	28	71	33	7	49	43	55	63	369
24	32	36	70	8	42	47	51	59	369
13	14	15	16	41	66	67	68	69	369
58	50	46	40	74	12	35	31	23	369
62	54	39	48	75	34	11	27	19	369
78	38	56	52	76	30	26	10	3	369
37	80	64	60	77	22	18	2	9	369
369	369	369	369	369	369	369	369	369	369

Computer Coding for 9×9 Squares

This part contains only the wheel portions Wheel 9(1) and Wheel 9(2) used for the last two magic squares.

(a) Wheel 9(1)

(b) Wheel 9(2)

This completes the Octagon A method. To go back to Part II. Go to Part IV. Go back to homepage.