NON-CONSECUTIVE MAGIC SQUARES WHEEL METHOD (Part IV)

A Discussion of the non-consecutive Magic Square Wheel Method

Construction of these magic squares requires is an approach which differs from the traditional step methods such as the Loubère and Méziriac. The magic square is constructed as follows using a complimentary table as a guide. The cells in the same color are associated with one another; for example, only 3 → 4 → 5 → 45 → 46 → 47 are still pair-associated consecutively together, while the other colored pairs are not. Because of the way these pairs are grouped the 7x7 square produced will not be initially magic but must be converted into one by a series of steps.

1. The left diagonal is filled with the group of numbers ½ (n²-n+2) to ½(n²+n) in consecutive order (top left corner to the right lower corner) from the numbers listed in the complementary table described above, for example using n = 7. For a 7x7 square the numbers in the left diagonal correspond to 22 → 23 → 24 → 25 → 26 → 27 → 28 (Square A1). Then add the right diagonal in reverse order from bottom left corner to the right upper corner choosing only from the pair {4,5,6,45,46,47}.


2. This is followed by the central column from the pairs {7,8,9,41,42,43} in regular order and then by the central row from the pairs {1,2,6,44,48,49} in reverse order (Square A2). We now have a partial square with not all sums equal to 175.


3. The result of these operations figuratively speaking resembles the hub and spokes of a wheel where the cells in color correspond to the spoke and hub of the wheel, with the non-consecutive pairs in different colors.

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A1 22                     46     23             45             24     44                     25                     6     26             5             27     4                     28

⇒

A2 22         7         46     23     8     45             24 9 44         49 48 44 25 6 2 1         6 41 26             5     42     27     4         43         28

⇒



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4. Fill in the spoke portions of the square with the twelve pairs that are left over (Square A3 and A4). This is done as follows:
   
   Use the small squares (from the complementary table) as an aid in in how to place the adjacent pair of numbers (in white), which may be placed in either a clockwise or anticlockwise manner as shown below:



10		40		10		40
↓	↗	↓		↑	↙	↑
40		39		40		39



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A3 175 22 10 12 7 38 40 47 176 14 23     8     46 35 126 16     24 9 45     33 127 49 48 44 25 6 2 1 175 34     5 41 26     17 123 36 4     42     27 15 124 3 39 37 43 13 11 28 174 174 124 122 175 128 126 176 175

⇒

A4 175 22 10 12 7 38 40 47 176 14 23 18 8 32 46 35 176 16 20 24 9 45 29 33 176 49 48 44 25 6 2 1 175 34 30 5 41 26 21 17 174 36 4 31 42 19 27 15 174 3 39 37 43 13 11 28 174 174 174 171 175 179 176 176 175

⇒



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5. At this point all sums are not equal to 175 so modifications are carried out to generate a magic square. Convert {7,8,9,41,42,43}, respectively, to {6,7,8,42,43,44}. This converts all sums in the last grey row to 175 (Square A5).


6. Also do this to the center row by converting {1,2,6,44,48,49} to {0,1,2,48,49,50}, respectively, which converts all sums in the last row to 175 (Square A6).


7. The complementary table for Square 6 is also shown with the different color cells where cells {9,41} are not part of this set.



A5 175 22 10 12 6 38 40 47 175 14 23 18 7 32 46 35 175 16 20 24 8 45 29 33 175 49 48 44 25 6 2 1 175 34 30 5 42 26 21 17 175 36 4 31 43 19 27 15 175 3 39 37 44 13 11 28 175 174 174 171 175 179 176 176 175

⇒

A6 175 22 10 12 6 38 40 47 175 14 23 18 7 32 46 35 175 16 20 24 8 45 29 33 175 50 49 48 25 2 1 0 175 34 30 5 42 26 21 17 175 36 4 31 43 19 27 15 175 3 39 37 44 13 11 28 175 175 175 175 175 175 175 175 175



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0	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
50	49	48	47	46	45	44	43	42	41	40	39	38	37	36	35	34	33	32	31	30	29	28	27	26



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This completes Part IV of the non-consecutive Magic Square Wheel method. Part V contains a third 7x7 variation.
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Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com