NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part W5

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 {104,108,112,116,120,61,2,6,10,14,18} and {20,16,12,8,60,61,62,114,110,106,102}

  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 5 up to the number 20, 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 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, 4 is added to 66 using numeric superscripts while in column 8, 33 is added to 37, followed by their complements, using alphabetic superscripts.
  5. Fill in similarly the internal 7x7 square (color cells) (Square A4). For example in row 3, 21 is added to 77 and 26 to 72. In column 9, 27 is added to 71 and 28 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, 23 is added to 84, 24 to 83 and 22 to 86 using numeric superscripts. In column 10, 31 is added to 76, 32 to 75 and 41 to 67 followed by their complements, using numeric superscripts.
  7. Finally fill in the external 11x11 square (color cells) (Square A6). For example in row 1, 25 is added to 78, 29 to 74, 34 to 87 and 42 to 79. In column 11, 30 is added to 73, 40 to 63, 53 to 68 and 57 to 64 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. 4 ... 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
    118 ... 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 858483
    531 ...251 151 161 162 201 252 253 254 202 203 163 164 5a 21 21 151 5a 161 162
    40 41 42 43 44 4546474849505152 53545556575859 60
    61
    8281807978 7776757473727170 696867666564 6362
    204 152 22 22 201 251 163 164 202 203 252 253 254 23 23 152 531 24 24 204
  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
104 17 102
  108 13 106
  112 9 110
  116 5 114
  120 1 62
191511 7 3 61 119 115111 107103
60 121 2
  8 117 6
  12 113 10
  16 109 14
20 105 18
A2 (Δ=1,δ=4)
104 17 102 448 2(103+121)
  108 13 106 322107x2+108
  112 9 110 19698x2
  116 5 114 7070
  120 1 62
191511 7 3 61 119 115111 107103
60 121 2
  8 117 6 174 174
  12 113 10 292 2x146
  16 109 14 402137x2+136
20 105 18 528 2(123+141)
528410292 174 70 196322448
A3
104 17 102
  108 13 106
  112 9 110
1164 5 66 114
89 120 1 62 33
191511 7 3 61 119 115111 107103
8560 121 2 37
8 118 117 56 6
  12 113 10
  16 109 14
20 105 18
A4
104 17 102
  108 13 106
112 21 26972 77 110
95 1164 5 66 114 27
9489 120 1 62 33 28
191511 7 3 61 119 115111 107103
528560 1212 37 70
518 118 117 56 6 71
12 101 96 113 5045 10
  16 109 14
20 105 18
A5
104 17 102
108 23 2422 13 86 8384 106
91 112 2126 972 77 110 31
9095 1164 5 66 114 27 32
8194 89 120 1 62 33 2841
191511 7 3 61 119 115111 107103
555285 60 121 2 37 7067
4751 8 118 117 56 6 71 75
46 12 101 96 113 5045 10 76
16 99 98100 109 3639 38 14
20 105 18
A6
10425 2934 4217 7987 7478 102
92108 23 2422 13 86 8384 10630
8291 112 2126 972 77 110 3140
699095 1164 5 66 114 27 3253
658194 89 120 1 62 33 2841 57
191511 7 3 61 119 115111 107103
58555285 60 121 2 37 706764
544751 8 118 117 56 6 71 7568
5946 12 101 96 113 5045 10 76 63
4916 99 98100 109 3639 38 1473
2097 9388 80105 4335 4844 18
A7
10425 29 3442 17 79 87 74 78 102
92108 23 2422 13 86 8384 10630
8291 112 2126 972 77 110 31 40
6990 95 1164 5 66 114 27 32 53
6581 9489 120 1 62 33 2841 57
1915 11 7 3 61 119 115111107 103
5855 5285 60 121 2 377067 64
54 47 518 118 117 566 7175 68
5946 12 101 96 113 5045 10 76 63
4916 99 98100 109 36 39 38 1473
20 97 93 88 80105 43 35 4844 18
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 41
 
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 W5 of a 11x11 Magic Square Wheel Spoke Shift method. To go to Part W6 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez