Magic Squares Wheel Method-Redux Part VI

Order 11^th Variants - Two Border Squares

The wheel method is a means of constructing magic squares by a random access means instead of sequentially like the Loubère and Méziriac methods which has been rewritten in a more simplified form. The method patially fills up a square to form a wheel structure using numbers chosen from a complementary table of order n then randomly fills up the rest of the square with numbers chosen from whatever is left in the complementary table. This paper is a simplification of the original paper taking a more facile and explanatory approach. Filling up the non spoke numbers can be compared to the filling up a large Soduku square but using complementary table and Summation tables to aid in the filling of the square which will be discussed below.

Two 11^th order variants are constructed so that the group of numbers in the left diagonal ½(n²-n+2) to ½(n²+n) line up according to the following reverse order 60 → 59 → 58 → 57 → 56 → 61 → 66 → 65 → 64 → 63 → 62. This order is necessary in order to generate border squares where every square is magic starting with the 3^rd up to the 11^th. The rest of the spoke numbers come from the first 12 pairs (in light brown) of the complementary table shown at the end. A pair corresponds to a number and its complement which are positioned on the square according to symmetrical considerations, i.e., each value from a pair of complements are equidistant from the center cell. In this case the pairs were chosen from the complementary table as follows: the first 4 pairs placed in the central row, the second 4 pairs placed on the right diagonal and the third 4 pairs in the central column all placed in random order, not sequentially, as was done in Part I. This section is a continuation of Part V which produced two 9^th order border squares.

1. Construct the wheel according to the method employed in Part I (Squares I-IV).
2. Aside I: If we sum up the numbers in each row (R) and column (C) of square IV we get the results in the Summation table where the sums can be broken down into the requisite sum pairs. These pairs can only take on the values of 81, 82 or 83, values which correspond to adjacent complement pairs. We then take these values, go over to the complement table and choose the complement pairs that fulfill this sum requirement.
3. Aside II: If we look at the parity column we can see that the parity of the two row or column numbers (R/C) in (1,9), (2,8), (3,7) and (4,6) must be the same (where O is odd and E is even) in order for the square to be magic. Therefore, having the summation table as a guide is a necessity in order to facilitate the placement of the complement numbers. It's also good to have the accompanying complement table on hand so as to add up and cross out the pair of numbers as they are chosen.
4. Fill in the 5x5 square, the 7x7, the 9x9 and finally the 11x11 according to the method used in Part IV.
5. This produces the border Square IV, where the Magic Sum of the 3^rd, 5^th, 7^th, 9^th and 11^th order squares is 183, 305, 427, 549 and 671, respectively.

Summation Table

R/C	Sum	Pair Sum	Parity
1	484	121+121+121+121	O+O+O+O
2	485	121+121+121+122	O+O+O+E
3	486	121+121+122+122	O+O+E+E
4	487	121+122+122+122	O+E+E+E
5	488	122+122+122+122	E+E+E+E
5	-	-	-
7	488	122+122+122+122	E+E+E+E
6	489	122+122+122+123	E+E+E+O
7	490	122+122+123+123	E+E+O+O
8	491	122+123+123+123	E+O+O+O
9	492	123+123+123+123	O+O+O+O

Square I 60 15 112 59 14 113 58 13 114 57 12 115 56 11 116 117 118 119 120 121 61 1 2 3 4 5 6 111 66 7 110 65 8 109 64 9 108 63 10 107 62

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Square II 60 16 15 105 112 59 18 14 103 113 58 32 20 13 101 89 114 34 57 22 12 99 115 88 24 26 28 30 56 11 116 92 94 96 98 117 118 119 120 121 61 1 2 3 4 5 97 95 93 91 6 111 66 31 29 27 25 87 7 100 110 23 65 35 8 90 102 109 21 33 64 9 104 108 19 63 10 106 107 17 62

Square III 60 16 15 105 112 59 36 38 18 14 103 83 85 113 40 58 32 20 13 101 89 114 82 42 34 57 22 12 99 115 88 80 24 26 28 30 56 11 116 92 94 96 98 117 118 119 120 121 61 1 2 3 4 5 97 95 93 91 6 111 66 31 29 27 25 79 87 7 100 110 23 65 35 43 81 8 90 102 109 21 33 64 41 9 86 84 104 108 19 39 37 63 10 106 107 17 62

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

6. The complementary table for square IV is shown below in two parts:

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	27	28	29	30	31
121	120	119	118	117	116	115	114	113	112	111	110	109	108	107	106	105	104	103	102	101	100	99	98	97	96	95	94	93	92	91

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

7. Instead of proceeding as above we can start by adding values to the first and last rows and columns to give (Square V) and then follow up by filling in the rest of the square by going inwards. This produces the second border square VI. Note the Summation table for this square is the same as the one above:

Square V 60 16 18 20 22 15 99 101 103 105 112 25 59 14 113 97 27 58 13 114 95 29 57 12 115 93 31 56 11 116 91 117 118 119 120 121 61 1 2 3 4 5 92 6 111 66 30 94 7 110 65 28 96 8 109 64 26 98 9 108 63 24 10 106 104 102 100 107 23 21 19 17 62

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

8. The complementary table for square VI is shown below:

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