NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part S4

Picture of a wheel

How to Spoke Shift 11x11 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).

In addition, the symbol Δ which has been used to specify a number added to or subtracted from the constants a, b or c in the first internal 3x3 magic square will always equal 1.


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 3132 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 5152 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, a new symbol δ specifies the difference between entries on the diagonals and center row and column where δ = 4 in all our cases except for the left diagonal of the internal 3x3 square (as shown below):

3x3 template
c+Δ a b+Δ
a+2Δ b c
b-Δ c+2Δ a+Δ

To avoid spaghetti type connections between paired non-spoke numbers, a coded system ( which I call "coded connectivity" as opposed to lined connectivity) employs a number and superscript 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 such as adding 21 to the complement of 31, i.e., 21 + 91, to give 112. While, 7a means that this number is added to a non-complementary number 7a both which are 7 units apart. In addition, if we look at the complementary table above 21 corresponds to the sum of 1 + 120, while 2-1 to the sum of 2 + 121. When either of the two sums is required, the ( ) or the (-) shows which one is being used.

A 11x11 Transposed Magic Square Using the Diagonals {120,116,112,108,104,61,18,14,10,6,2} and {4,8,12,16,60,61,62,106,110,114,118}

  1. Add one to the first row center of a 11x11 square, 2 to the rightmost bottom cell, 3 to the center of the first column and 4 to the leftmost bottom cell. Repeat (i.e. spiraling towards the center) for the next 18 numbers, followed by their complementary numbers (Square A1).
  2. Add the numbers 60 and 62 to the empty two internal cells. This generates a 3x3 internal magic square (Square 2).
  3. Sum up the empty 1st row, the empty 11th; the 3rd, the 10th; the 3nd;the 9th and the 4nd;the 8th rows. Do the same for the columns (green cells). The values are in the twelve column and are equal to the multiplied values in the thirteenth column. That is including both colums and rows, there should be 12 sums which add up to 104; four that add up to 98 and two of 78.
  4. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square 3). For example 35 is added to 43 and 36 to 42, followed by their complements, using alphabetic superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square 4). For example 21 is added to 77 and 22 to 76 in the row. Add 28 to 70 and 29 to 69 in the columns followed by their complements, using alphabetic superscripts.
  6. Fill in the internal 9x9 square (color cells) (Square 5). For example (50 is added to 58, 51 to 57 using alphabetic superscripts) and 31 to 67 in the row. Add 30 to 74, 44 to 68 and 32 to 66 in the column followed by their complements, using numeric superscripts.
  7. Finally fill in the external 11x11 square (color cells) (Square 6). For example using either alphabetic or numeric superscripts 20 is added to 88, 23 to 85, 24 to 4 and 25 to 83 in the row. Also add 26 to 82, 27 to 1, 33 to 75 and 49 to 59 in the column followed by their complements.
  8. No connectivity between numbers using complicated spaghetti lines will be attempted but will now be replaced by the coded table and by the color complementary table shown at the end.
  9. Below is the coded connections to this square:
  10. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
    102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79
    151 251 252 152 153 154 155 156 253 254 191 255 256 157 151 9a 7a 152 153 154 155 156 7a 9a
    44 45 46 47 48 49 50 51 52 53 54 55 56 5758 59 60
    61
    78 77 7675 74 73 72 71 70 69 68 67 66 65 64 63 62
    111 251 252 157 191 11a 9b 7b 253 254 111 255 256 7b 9b 11a
  11. The coded method as well as the color squares (below Square 7) will be used from here on due too many lines crossing and crisscrossing.
  12. Square A6 shows the 5 border squares in "border format".
  13. The complement table below also shows how the color pairs are layed out (for comparison with Square A6).
A1
120 1 118
  116 5 114
  112 9 110
  108 13 106
  104 17
3711 15 19 61 103 107111 115119
105 18
  16 109 14
  12 113 10
  8 117 6
4 121 2
A2 (Δ=1,δ=4)
120 1 118 432 108x4
  116 5 114 314108x2 + 98
  112 9 110 19698x2
  108 13 106 7878
  104 17 62
3 711 15 19 61 103 107111115 119
60 105 18
  16 109 14 166
  12 113 10 292
  8 117 6 418
4 121 2544
544418292 166 78 196314432 314104+112+98
A3
120 1 118
  116 5 144
  112 9 110
10835 13 43 106
86 104 17 62 36
3 711 15 19 61 103 107111115 119
8060 105 18 42
16 87 109 79 14
  12 113 10
  8 117 6
4 121 2
A4
120 1 118
  116 5 144
112 21 22976 77 110
94 10835 13 43 106 28
9386 104 17 62 36 29
3 711 15 19 61 103 107111115 119
538060 105 18 42 69
5216 87 109 79 14 70
12 101 100 113 4645 10
  8 117 6
4 121 2
A5
120 1 118
116 50 5131 5 67 57 58 114
92 112 2122 976 77 110 30
7894 10835 13 43 106 28 44
9093 86 104 17 62 36 2932
3 711 15 19 61 103 107111115 119
565380 60 105 18 42 6966
5452 16 87 109 79 14 70 68
48 12 101 100 113 4645 10 74
8 72 7191 117 5565 64 6
4 121 2
A6
12020 2324 251 8384 8588 118
96116 50 5131 5 67 5758 11426
9592 112 2122 976 77 110 3027
8978 94 10835 13 43 106 28 4433
739093 86 104 17 62 36 2932 49
3 711 15 19 61 103 107111115 119
635653 80 60 105 18 42 696659
475452 16 87 109 79 14 70 6875
4148 12 101 100 113 4645 10 7481
408 72 7191 117 55 6564 6 82
4102 9998 97121 3938 3734 2
A7
12020 23 2425 1 83 84 85 88 118
96116 50 5131 5 67 5758 11426
9592 112 2122 976 77 110 30 27
8978 94 10835 13 43 106 28 44 33
7390 9386 104 17 62 36 2932 49
37 11 15 19 61 103 107111115 119
6356 5380 60 105 18 426966 59
47 54 5216 87 109 79 14 70 68 75
4148 12 101 100 113 4645 10 74 81
408 72 7191 117 55 65 64 682
4 102 99 98 97121 39 3837 34 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
 
121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103
20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40 41
 
102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81
42 43 44 4546 4748 49 5051 5253 54 5556 57 5859 60
61
8079 7877 76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part S4 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com