NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part T3

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 {44,48,52,56,60,61,62,66,70,74,78}

  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 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 118.
  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 114 while in column 8, 12 is added to 106, 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 98 and 21 to 97. In column 9, 22 is added to 96 and 23 to 95 followed by their complements, using numeric superscripts.
  6. Fill in similarly the internal 9x9 square (color cells) (Square 5). For example in row 2, 28 is added to 90, 29 to 89 and 30 to 88 using alphabetic superscripts. In column 10, 31 is added to 87, 36 to 83 and 37 to 81 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) 38 is added to 80, 39 to 79, 45 to 73 and 46 to 72. In column 11, 47 is added to 71, 53 to 65, 54 to 64 and 55 to 63 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 ... 8 ... 12 ... 16 ... 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
    118 ... 114 ... 110 ... 106 ... 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87
    51 ... 51 ... 52 ... 52 ... 53 54 55 56 53 54 55 56 57 58 59 510 57 58 59 510
    36 3738 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 5758 59 60
    61
    86 85 84 83 82 81 80 79 78 77 7675 74 73 72 71 70 69 68 67 66 65 64 63 62
    511 512 513 514 511 512 513 514 ... 515 516 517 ... 515 516 517 ... 518 519 520 ... 518 519 520
  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 78
  116 5 74
  112 9 70
  108 13 76
  104 17 62
3711 15 19 61 103 107111 115119
60 105 18
  56 109 14
  52 113 10
  48 117 6
44 121 2
A2 (Δ=1,δ=4)
120 1 78 472 118x4
  116 5 74 354118x3
  112 9 70 236118x2
  108 13 66 118118
  104 17 62
3 711 15 19 61 103 107111115 119
60 105 18
  56 109 14 126 126
  52 113 10 252 126x2
  48 117 6 378126x3
44 121 2504 126x4
504378252 126 118 236354472
A3
120 1 78
  116 5 74
  112 9 70
1084 13 114 66
110 104 17 62 12
3 711 15 19 61 103 107111115 119
1660 105 18 106
56 118 109 8 14
  52 113 10
  48 117 6
44 121 2
A4
120 1 78
  116 5 74
112 20 21997 98 70
100 1084 13 114 66 22
99110 104 17 62 12 23
3 711 15 19 61 103 107111115 119
271660 105 18 106 95
2656 118 109 8 14 96
52 102 101 113 2524 10
  48 117 6
44 121 2
A5
120 1 78
116 28 2930 5 88 89 90 74
91 112 2021 997 98 70 31
86100 1084 13 114 66 22 36
8599 110 104 17 62 12 2337
3 711 15 19 61 103 107111115 119
412716 60 105 18 106 9581
4026 56 118 109 8 14 96 82
35 52 102 101 113 2524 10 87
48 94 9392 117 3433 32 6
44 121 2
A6
12038 3945 461 7273 7980 78
75116 28 2930 5 88 8990 7447
6991 112 2021 997 98 70 3153
6886 100 1084 13 114 66 22 3654
678599 110 104 17 62 12 2337 55
3 711 15 19 61 103 107111115 119
594127 16 60 105 18 106 958163
584026 56 118 109 8 14 96 8264
5735 52 102 101 113 2524 10 8765
5148 94 9392 117 34 3332 6 71
4484 8377 76121 5049 4342 2
A7
12038 39 4546 1 72 73 79 80 78
75116 28 2930 5 88 8990 7447
6991 112 2021 997 98 70 31 53
6886 100 1084 13 114 66 22 36 54
6785 99110 104 17 62 12 2337 55
37 11 15 19 61 103 107111115 119
5941 2716 60 105 18 1069581 63
58 40 2656 118 109 8 14 96 82 64
5735 52 102 101 113 2524 10 87 65
5148 94 9392 117 34 33 32 671
44 84 83 77 76121 50 49 43 42 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 T3 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez