NON-CONSECUTIVE MAGIC SQUARES WHEEL METHOD (Part V)

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. In addition, on this page we must perform a double modification to arrive at a magic partially non-consecutive square.

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 {4,5,7,43,45,46}.
2. This is followed by the central column from the pairs {2,3,6,44,47,48} in regular order and then by the central row from the pairs {1,8,9,41,42,49} 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.

A1 22 46 23 45 24 43 25 7 26 5 27 4 28

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A2 22 2 46 23 3 45 24 6 43 49 42 41 25 9 8 1 7 44 26 5 47 27 4 48 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

A3 175 22 10 12 2 38 40 46 170 14 23 3 45 35 121 16 24 6 43 33 122 49 42 41 25 9 8 1 175 34 7 44 26 17 128 36 5 47 27 15 130 4 39 37 48 13 11 28 180 175 119 121 175 129 131 175 175

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A4 175 22 10 12 2 38 40 46 170 14 23 18 3 32 45 35 170 16 20 24 6 43 29 33 171 49 42 41 25 9 8 1 175 34 30 7 44 26 21 17 179 36 5 31 47 19 27 15 180 4 39 37 48 13 11 28 180 175 169 170 175 180 181 175 175

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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 {2,3,6,44,47,48}, respectively, to {7,8,10,40,42,43}. This converts all sums in the last grey row to 175 (Square A5).
6. Also do this to the center row by converting {42,41,9,8} to {48,46,4,2}, respectively, which converts all sums in the last row to 175 (Square A6).

A5 175 22 10 12 7 38 40 46 175 14 23 18 8 32 45 35 175 16 20 24 10 43 29 33 175 49 42 41 25 9 8 1 175 34 30 7 40 26 21 17 175 36 5 31 42 19 27 15 175 4 39 37 43 13 11 28 175 175 169 170 175 180 181 175 175

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A6 175 22 10 12 7 38 40 46 175 14 23 18 8 32 45 35 175 16 20 24 10 43 29 33 175 49 48 46 25 4 2 1 175 34 30 7 40 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

7. At this point 6 duplicates are present {4,7,10,40,43,46}. Four can be removed first followed by the last two. For example, by adding n² = 49 to each of {4,7,10,16,27,35,43} square A7 is obtained.
8. Adding n² = 49 to each of {14,20,37,40,46,53,76} removes the other two duplicates to give Square A8 with the last modifications in yellow where S = 273 and S = ½(n³ + 29n).
9. 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.

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A7 224 22 59 12 7 38 40 46 224 14 23 18 8 32 45 84 224 65 20 24 10 43 29 33 224 49 48 46 25 53 2 1 224 34 30 56 40 26 21 17 224 36 5 31 42 19 76 15 224 4 39 37 92 13 11 28 224 224 224 224 224 224 224 224 224

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A8 273 22 59 12 7 38 40 95 273 63 23 18 8 32 45 84 273 65 69 24 10 43 29 33 273 49 48 46 25 102 2 1 273 34 30 56 89 26 21 17 273 36 5 31 42 19 125 15 273 4 39 86 92 13 11 28 273 273 273 273 273 273 273 273 273

10. In addition, the modified complementary table is depicted below with the modified cell numbers in yellow and green where some of the numbers no longer show complementarity. Also note that 7, 10 40 and 46, not shown on the complementary table because of duplication, are present in Square A8.

This completes Part V of the non-consecutive Magic Square Wheel method. To go back to homepage.