NON-CONSECUTIVE MAGIC SQUARES WHEEL METHOD (Part III)

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 2 → 3 → 7 → 43 → 47 → 48 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).


2. 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}.


3. This is followed by the central column from the pairs {2,3,7} in regular order and then by the central row from the pairs {1,6} in reverse order (Square A2). We now have a partial square with not all sums equal to 175.


4. 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         2         46     23     3     45             24 7 44         49 42 41 25 9 8 1         6 43 26             5     47     27     4         48         28

⇒



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5. 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 2 38 40 46 170 14 23     3     45 35 120 16     24 7 44     33 124 49 42 41 25 9 8 1 175 34     6 43 26     17 126 36 5     47     27 15 130 4 39 37 48 13 11 28 180 175 119 120 175 130 131 175 175

⇒

A4 175 22 10 12 2 38 40 46 170 14 23 18 3 32 45 35 170 16 20 24 7 44 29 33 173 49 42 41 25 9 8 1 175 34 30 6 43 26 21 17 177 36 5 31 47 19 27 15 180 4 39 37 48 13 11 28 180 175 169 169 175 181 181 175 175

⇒



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6. At this point all sums are not equal to 175 so modifications are carried out to generate a magic square. Convert 2, 3 and 7, respectively, to 7, 8 and 9. Also convert the complements to those of the latter. This converts all sums in the last grey row to 175 (Square A5).


7. Also do this to the center row by converting 8 and 9 to 2 and 3, respectively, (the same to the complements) which converts all sums in the last row to 175 (Square A6).


8. The complementary table for Square 6 is also shown with the different color cells.



A5 175 22 10 12 7 38 40 46 175 14 23 18 8 32 45 35 175 16 20 24 9 44 29 33 175 49 42 41 25 9 8 1 175 34 30 6 41 26 21 17 175 36 5 31 42 19 27 15 175 4 39 37 43 13 11 28 175 175 169 169 175 181 181 175 175

⇒

A6 175 22 10 12 7 38 40 46 175 14 23 18 8 32 45 35 175 16 20 24 9 44 29 33 175 49 48 47 25 3 2 1 175 34 30 6 41 26 21 17 175 36 5 31 42 19 27 15 175 4 39 37 43 13 11 28 175 175 175 175 175 175 175 175 175



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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
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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9. Although Square A6 is simple to produce retaining the sum of 175, Square A4 can be converted to a different type (one of many) by first converting the appropriate cells in the middle row to Square A7.


10. Convert the cells in the set {2,18,7,43,31,48} to {7,23,9,41,26,43} giving the magic square A8 with sum = 175.


11. However, since 3, 23 and 26 are duplicates, add n² = 49 to the the cells in the set {3,4,7,21,23,26,33} to produce the square A9 with S = 224 and S = ½(n³ + 15n), and with the modified cells in yellow.


12. The last equation takes into account that the sum has been modified from the known equation S = ½(n³ + n) to the general equation as was shown in:
    
    S = ½(n³ ± an)

    which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or - values. For example when a = 1 the normal magic sum S is implied. When a takes on different odd values S gives the magic sum of a modified magic square. It can be seen from the equation that addition or subtraction of n² to some of the cells in the square gives rise to a new magic square, which in this case have cell numbers greater than 49.

This completes Part III of the non-consecutive Magic Square Wheel method. Part IV contains a second 7x7 variation.
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