NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part U3

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 = 8 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 {120,112,104,96,88,61,34,26,18,10,2} and {28,36,44,52,60,61,62,70,78,86,94}

  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 27 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 1).
  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 20 sums which add up to 114. However, since four sums of 114 cannot be generated the sums of 228 will be added as 104 + 124.
  4. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square 3). For example in row 4, 4 is added to 110 while in column 8, 5 is added to 109, followed by their complements, using numeric superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square 4). For example in row 3, 20 is added to 84 and 39 to 85. In column 9, 40 is added to 64 and 59 to 65, followed by their complements, using numeric superscripts.
  6. Fill in similarly the internal 9x9 square (color cells) (Square 5). For example in row 2, 6 is added to 108, 7 to 107 and 8 to 106 using numeric superscripts. In column 10, 21 is added to 93, 22 to 92 and 23 to 91 followed by their complements, using numeric superscripts.
  7. Finally fill in the external 11x11 square (color cells) (Square 6). For example in row 1, (using numeric superscripts) 24 is added to 90, 41 to 73, 42 to 72 and 43 to 71. In column 11, 45 is added to 69, 46 to 68, 47 to 67 and 48 to 66 along with 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 where the colored "spoke"cells are not included in the coding:
  10. 4 5 6 7 8 ... 12 13141516 ... 20 21 22 2324... 29 30 31 32 ...
    118 117 116 115114 ... 110 109108107106 ... 102 101 1009998...93 92 91 90 ...
    131 132 133 134 135 136 137 138 ... 131 132 133 134 135 136 137 138 ... 191 96 97 98 99 ... 96 97 98 99 ...
    37 383940414243444546 4748495051525354 55565758 5960
    61
    85848382818079787776 7574737271706968 676665646362
    31 191 31 192 910 911 912 ... 913 914 915 916 910 911 912... 913 914 915 916 32 192 32
  11. The coded method as well as the color squares (below Square 7) will be used from here on due to excessive crisscrossing of lines.
  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 94
  112 9 86
  104 17 78
  96 25 70
  88 33 62
31119 27 35 61 87 95103 111119
60 89 34
  52 97 26
  44 105 18
  36 113 10
28 121 2
A2 (Δ=1,δ=8)
120 1 94 486 114x4
  112 9 86 342114x3
  104 17 78 228104+124
  96 25 70 114114
  88 33 62
31119 27 35 61 87 95103 111119
60 89 34
  52 97 26 130 130
  44 105 18 260 140+120
  36 113 10 390126x3
28 121 2520 114x4
520390260 130 114 228342456
A3
120 1 94
  112 9 86
  104 17 78
964 25 110 70
117 88 33 62 5
31119 27 35 61 87 95103 111119
1360 89 34 109
52 118 97 12 26
  44 105 18
  36 113 10
28 121 2
A4
120 1 94
  112 9 86
104 20 391785 84 78
82 964 25 110 70 40
63117 88 33 62 5 59
31119 27 35 61 87 95103 111119
571360 89 34 109 65
5852 118 97 12 26 64
44 102 83 105 3738 18
  36 113 10
28 121 2
A5
120 1 94
112 6 78 9 106 107108 86
101 104 2039 1785 84 78 21
10082 964 25 110 70 40 22
9963 117 88 33 62 5 5923
31119 27 35 61 87 95103 111119
315713 60 89 34 109 6591
3058 52 118 97 12 26 64 92
29 44 102 83 105 3738 18 93
36 116 115114 113 1615 14 10
28 121 2
A6
12024 4142 431 7172 7390 94
77112 6 78 9 106 107108 8645
76101 104 2039 1785 84 78 2146
7510082 964 25 110 70 40 2247
749963 117 88 33 62 5 5923 48
31119 27 35 61 87 95103 111119
563157 1360 89 34 109 659166
553058 52 118 97 12 26 64 9267
5429 44 102 83 105 3738 18 93 68
5336 116 115114 113 1615 14 1069
2898 8180 79121 5150 4932 2
A7
12024 41 4243 1 71 72 73 90 94
77112 6 78 9 106 107108 8645
76101 104 2039 1785 84 78 21 46
75100 82 964 25 110 70 40 22 47
7499 63117 88 33 62 5 5923 48
311 19 27 35 61 87 95103111 119
5631 5713 60 89 34 1096591 66
55 30 5852 118 97 1226 6492 67
5429 44 102 83 105 3738 18 93 68
5336 116 115114 113 16 15 14 1069
28 98 81 80 79121 51 50 4932 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 U3 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez