NEW MAGIC SQUARES WHEEL METHOD - BORDER SQUARES

Part X4

How to generate 11x11 Border Magic Squares

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).

This site introduces a new methods used for the construction of border wheel type squares except that the initial spoke parts are added in a somewhat different manner than in the original wheel method. The method consists of forming an internal 3x3 magic square, then generating all subsequent border magic squares. as was done in the original method. The difference between this type of square and the original is that the numbers (and complements) are added consecutively, starting from 1, at the center top cell. Subsequent numbers are added to each of the diagonals and the center row. The left diagonal of the internal 3x4 square, however, deviates from this arrangement where the three numbers on this diagonal are ½(n² − 1), ½(n² + 1), ½(n² + 3).

This site introduces a new method used for the construction of border wheel type squares. The method consists of forming a 7x7 internal Wheel magic square then filling in the external 1,2 and 10,11 rows and columns with the requisite non-spoke numbers as will be shown below.

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40

101	100	99	98	97	96	95	94	93	92	91	90	89	48	87	86	85	84	83	82

41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																				61
81	80	79	78	77	76	75	74	73	72	71	60	69	68	67	66	65	64	63	62

Furthermore, the symbol δ (where δ = 4) specifies the difference between entries on the diagonals and center row and column not situated on the center 7x7 square (shown in square A1)

The non-spoke entries in rows and columns 1,2 and 10,11 are added according to a coded system ( which I call "coded connectivity" as opposed to lined connectivity) employing a number and superscript and where the number gives the difference between two paired numbers and the superscript shows which two numbers are paired together. For example, 11¹ says that this number is added to a second complementary number 11¹ separated by a distance of 11. From the complementary table above 1 + 71 is such an example. While, 7^a means that this number is added to a non-complementary number 7^a both which are 7 units apart. In addition, if we look at the complementary table above 2¹ corresponds to the sum of 1 + 80, while 2^-1 to the sum of 2 + 81. When either of the two sums is required, the number is preceded by either a ( ) or by (-).

A 11x11 Transposed Magic Square Using the Diagonals {32,36,58,59,60,61,62,63,64,86,90} and {30,34,56,39,78,61,44,83,66,88,92}

Fill in the internal 7x7 square with the numbers 37 through 85 according to the wheel method to form a magic square.
Subtract δ = 4 from 37 and add this number (33) to the center cell of row 2 and repeat again (33-4) = 29 and place this number in the center cell of row 1. Starting with the number 29 place consecutive numbers as shown in Square A1 in a spiraling fashion up to the number 36, followed by their complementary numbers (Square A1). This completes the spokes for the square, along with the nonspoke numbers of the 7x7 square.
Thus the spokes of the wheel are shown as follows: Left diagonal 32,36...86,90; right diagonal 30,34...88,92; central column 29,33...89,93; central row 31,35...87,91. (...) denotes the numbers from the 7x7 square (Square A1).
Sum up the rows with the diagonal and central row or column and subtract from 549 (sum of 9x9 internal square), to give the amounts required to complete the 9x9 square. The 13^th rows shows the numbers required. (Square A2).
Fill in the required pairs for the row and columns chosen from the coded table below coded in numeric superscripts.
Repeat for rows and columns 1,11. However, subtract the numbers from 671 (the sum of a 11x11 magic square). Square A2 shows 4 pairs of numbers are required and these are listed down. The coded table below shows which numbers pair up. Note that the sum of the numbers in rows and columns 1 and 11 sum up to 122 (Square A4).
Below is the coded connections to this square where the colored "spoke" cells and those labeled (...) i.e., the 7x7 entries, are not included in the coding.

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106


26¹	9¹	26²	9²	9³	9⁴	9⁵	9⁶	3¹	9¹	3¹	9²	9³	9⁴	9⁵	9⁶

17	18	19	20	21	22	23	24	25	26	27	28	...	58	59	60
																61
105	104	103	102	101	100	99	98	97	96	95	94	...	64	63	62


3²	3³	3²	3³	2¹	2¹	2²	2²	3⁴	26¹	3⁴	26²

Square A5 shows the 3 border squares in "border format".
The complementary table below also shows how the color pairs of Square A4 are layed out.

A1
92					29					90
	88				33				86
		82	50	53	37	71	70	64
		54	81	74	38	49	63	68
		57	48	80	39	62	76	65
31	35	43	44	45	61	77	78	79	87	91
		67	73	60	83	42	47	55
		66	59	46	84	75	41	56
		58	72	69	85	51	52	40
	36				89				34
32					93					30

⇒

A2
92					29					90	460	123+97+(120x2)
	88				33				86		342	114x3
		82	50	53	37	71	70	64
		54	81	74	38	49	63	68
		57	48	80	39	62	76	65
31	35	43	44	45	61	77	78	79	87	91
		67	73	60	83	42	47	55
		66	59	46	84	75	41	56
		58	72	69	85	51	52	40
	36				89				34		390	130x3
32					93					30	516	121+(121x2)+147
516	390								342	460

⇒

A3
92					29					90
	88	2	4	5	33	109	110	112	86
	116	82	50	53	37	71	70	64	6
	115	54	81	74	38	49	63	68	7
	114	57	48	80	39	62	76	65	8
31	35	43	44	45	61	77	78	79	87	91
	16	67	73	60	83	42	47	55	106
	15	66	59	46	84	75	41	56	107
	14	58	72	69	85	51	52	40	108
	36	120	118	117	89	13	12	10	34
32					93					30

⇒

A4
92	22	9	17	1	29	96	103	111	101	90
98	88	2	4	5	33	109	110	112	86	24
104	116	82	50	53	37	71	70	64	6	18
97	115	54	81	74	38	49	63	68	7	25
119	114	57	48	80	39	62	76	65	8	3
31	35	43	44	45	61	77	78	79	87	91
28	16	67	73	60	83	42	47	55	106	94
27	15	66	59	46	84	75	41	56	107	95
20	14	58	72	69	85	51	52	40	108	102
23	36	120	118	117	89	13	12	10	34	99
32	100	113	105	121	93	26	19	11	21	30

⇒

A5 Partial Border
92	22	9	17	1	29	96	103	111	101	90
98	88	2	4	5	33	109	110	112	86	24
104	116	82	50	53	37	71	70	64	6	18
97	115	54	81	74	38	49	63	68	7	25
119	114	57	48	80	39	62	76	65	8	3
31	35	43	44	45	61	77	78	79	87	91
28	16	67	73	60	83	42	47	55	106	94
27	15	66	59	46	84	75	41	56	107	95
20	14	58	72	69	85	51	52	40	108	102
23	36	120	118	117	89	13	12	10	34	99
32	100	113	105	121	93	26	19	11	21	30

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

101	100	99	98	97	96	95	94	93	92	91	90	89	88	87	86	85	84	83	82	81

42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																			61
80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

This completes the Part X4 of a 11x11 Magic Square Wheel Spoke Shift method.
Go back to homepage.

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40

101	100	99	98	97	96	95	94	93	92	91	90	89	48	87	86	85	84	83	82

41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																				61
81	80	79	78	77	76	75	74	73	72	71	60	69	68	67	66	65	64	63	62

92	22	9	17	1	29	96	103	111	101	90
98	88	2	4	5	33	109	110	112	86	24
104	116	82	50	53	37	71	70	64	6	18
97	115	54	81	74	38	49	63	68	7	25
119	114	57	48	80	39	62	76	65	8	3
31	35	43	44	45	61	77	78	79	87	91
28	16	67	73	60	83	42	47	55	106	94
27	15	66	59	46	84	75	41	56	107	95
20	14	58	72	69	85	51	52	40	108	102
23	36	120	118	117	89	13	12	10	34	99
32	100	113	105	121	93	26	19	11	21	30

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

101	100	99	98	97	96	95	94	93	92	91	90	89	88	87	86	85	84	83	82	81

42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																			61
80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40

101	100	99	98	97	96	95	94	93	92	91	90	89	48	87	86	85	84	83	82

41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																				61
81	80	79	78	77	76	75	74	73	72	71	60	69	68	67	66	65	64	63	62

92	22	9	17	1	29	96	103	111	101	90
98	88	2	4	5	33	109	110	112	86	24
104	116	82	50	53	37	71	70	64	6	18
97	115	54	81	74	38	49	63	68	7	25
119	114	57	48	80	39	62	76	65	8	3
31	35	43	44	45	61	77	78	79	87	91
28	16	67	73	60	83	42	47	55	106	94
27	15	66	59	46	84	75	41	56	107	95
20	14	58	72	69	85	51	52	40	108	102
23	36	120	118	117	89	13	12	10	34	99
32	100	113	105	121	93	26	19	11	21	30

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

101	100	99	98	97	96	95	94	93	92	91	90	89	88	87	86	85	84	83	82	81

42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																			61
80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40

101	100	99	98	97	96	95	94	93	92	91	90	89	48	87	86	85	84	83	82

41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																				61
81	80	79	78	77	76	75	74	73	72	71	60	69	68	67	66	65	64	63	62

92	22	9	17	1	29	96	103	111	101	90
98	88	2	4	5	33	109	110	112	86	24
104	116	82	50	53	37	71	70	64	6	18
97	115	54	81	74	38	49	63	68	7	25
119	114	57	48	80	39	62	76	65	8	3
31	35	43	44	45	61	77	78	79	87	91
28	16	67	73	60	83	42	47	55	106	94
27	15	66	59	46	84	75	41	56	107	95
20	14	58	72	69	85	51	52	40	108	102
23	36	120	118	117	89	13	12	10	34	99
32	100	113	105	121	93	26	19	11	21	30

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

121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102

21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

101	100	99	98	97	96	95	94	93	92	91	90	89	88	87	86	85	84	83	82	81

42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60
																			61
80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62