NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part W6

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.

This page unlike a previous method employs an internal 3x3 square containing the consecutive numerals 1, 2 and 3. In addition, the numerals for the 5x5, 7x7 and 9x9 squares are incrementally added starting with the number 5 and increasing to the number 20. See Square A1 below. The rest of the square is filled out as follows:

A 11x11 Transposed Magic Square Using the Diagonals {105,109,113,117,120,61,2,5,9,13,17} and {19,15,11,7,60,61,62,115,111,107,103}

  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 outwards from the center) starting with the number 4 up to the number 19, followed by their complementary numbers (Square A1). This differs from the previous square where the starting number was 5 and the final number 20.
  2. Add the numbers 60 and 62 to the empty two internal cells. This generates a 3x3 internal magic square (Square A1).
  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 twelfth column and are equal to the multiplied values in the thirteenth column. That is including both colums and rows, there should be 8 sums whose values are listed in column 12 (Square A2).
  4. Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square A3). For example in row 4, 34 is added to 35 while in column 8, 31 is added to 38, followed by their complements, both using alphabetic superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square A4). For example in row 3, 22 is added to 76 and 26 to 71. In column 9, 23 is added to 75 and 27 to 70, followed by their complements, using numeric superscripts.
  6. Fill in similarly the internal 9x9 square (color cells) (Square A5). For example in row 2, 20 is added to 73, 32 to 89 and 24 to 83 using numeric superscripts. In column 10, 21 is added to 72, 57 to 64 and 25 to 82 followed by their complements, using numeric superscripts.
  7. Finally fill in the external 11x11 square (color cells) (Square A6). For example in row 1, 30 is added to 81, 42 to 69, 43 to 68 and 28 to 86. In column 11, 44 is added to 67, 45 to 66, 48 to 63 and 29 to 85 along with their complements using numeric superscripts.
  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. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
    102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85848382
    301 302 251 252 161 162 261 262 91 92 121 8a 21 21 2a 2a 91 92 8a 161 162
    41 42 43 44 4546474849505152 53545556575859 60
    61
    81807978 7776757473727170 696867666564 6362
    121 122 123 124 125 251 252 126 301 302 261 262 122 123 124 125 22 22 126
  11. The coded method as well as the color squares (below Square A7) 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
105 16 103
  109 12 107
  113 8 111
  117 4 115
  120 1 62
181410 6 3 61 119 116112 108104
60 121 2
  7 118 5
  11 114 9
  15 110 13
19 106 17
A2 (Δ=1,δ=4)
105 16 103 447 3(111)+114
  109 12 107 32193+121+107
  113 8 111 19598+97
  117 4 115 6969
  120 1 62
181410 6 3 61 119 116112 108104
60 121 2
  7 118 5 175 175
  11 114 9 293 146+147
  15 110 13 411151+123+137
19 106 17 528 3(133)+130
529411293 175 69 195321447
A3
105 16 103
  109 12 107
  113 8 111
11734 4 35 115
91 120 1 62 31
181410 6 3 61 119 116112 108104
8460 121 2 38
7 88 118 87 5
  11 114 9
  15 110 13
19 106 17
A4
105 16 103
  109 12 107
113 22 26871 76 111
99 11734 4 35 115 23
9591 120 1 62 31 27
181410 6 3 61 119 116112 108104
528460 1212 38 70
477 88 118 87 5 75
11 100 96 114 5146 9
  15 110 13
19 106 17
A5
105 16 103
109 20 3224 12 83 8973 107
101 113 2226 871 76 111 21
6599 11734 4 35 115 23 57
9795 91 120 1 62 31 2725
181410 6 3 61 119 116112 108104
405284 60 121 2 38 7082
5847 7 88 118 87 5 75 64
50 11 100 96 114 5146 9 72
15 102 9098 110 3933 49 13
19 106 17
A6
10530 4243 2816 8668 6981 103
78109 20 3224 12 83 8973 10744
77101 113 2226 871 76 111 2145
746599 11734 4 35 115 23 5748
939795 91 120 1 62 31 2725 29
181410 6 3 61 119 116112 108104
374052 84 60 121 2 38 708285
595847 7 88 118 87 5 75 6463
5650 11 100 96 114 5146 9 72 66
5515 102 9098 110 3933 49 1367
1992 8079 94106 3654 5341 17
A7
10530 42 4328 16 86 68 69 81 103
78109 20 3224 12 83 8973 10744
77101 113 2226 871 76 111 21 45
7465 99 11734 4 35 115 23 57 48
9397 9591 120 1 62 31 2725 29
1814 10 6 3 61 119 116112108 104
3740 5284 60 121 2 387082 85
59 58 477 88 118 875 7564 63
5650 11 100 96 114 5146 9 72 66
5515 102 9098 110 39 33 49 1367
19 92 80 79 94106 36 54 5341 17
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 W6 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2015 by Eddie N Gutierrez