NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

BORDER and SPOKE SHIFT Part A7

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 left diagonal numbers don't have to be chosen from the consecutive group ½(n2-n+2) to ½(n2+n) but may be chosen from any other consecutive group of numbers. However, every spoke on the wheel consists of consecutive numbers and their complements. For example, instead of choosing the complementary numbers (56,57,58,59,60,61,62,63,64,65,66) from the list:

The new magic squares with n = 11 are constructed as follows using a complimentary table as a guide.

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

as the left diagonal another set (50,49,48,47,46,61,76,75,77,73,72) may be chosen as its replacement and the magic square is no longer dependent on ½(n2-n+2) to ½(n2+n) for the main diagonal. In addition, the square will be rotated 90° to the left to allow the lowest numbers to occupy the top of the central column. The symbol Δ will be used to specify a number added to or subtracted from the constants a, b or c to generate the first internal 3x3 magic square. In this case Δ is, consequently, obtained from the first number of the set (50,49,48,47,46), i.e., Δ = 61 − 46 in order to to generate the left diagonal,

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

A 9x9 Transposed Magic Square Using the Diagonals {102,103,104,104,106,61,16,17,18,19,20} and {50,49,48,47,46,61,76,75,77,73,72}

  1. Generate a 3x3 square using Δ=15, b=61 and a=1. (Square A1).
  2. Generate Square A2 by adding consecutive numbers to the two diagonals and the central column and row (the "spoke") and include their complements from the complement list above.
  3. To begin fill up the square add up the entries on the first row and subtract from 671 (the magic sum for a 11x11 square). This affords the value 492 which divided by 4 gives the sum of pairs needed to fill up that line, which in this case is (123 x 4). See Figure A.
  4. Repeat for row 2 except subtract the value from 549 (the magic sum for a 9x9 internal square). This gives a value of 123x3.
  5. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 123x2.
  6. Repeat for row 4 except subtract the value from 305 (the magic sum for a 5x5 internal square). This gives a value of 123.
  7. Do the same for rows 8, 9 10 and 11 obtaining, respectively, 484, 363, 242 and 121.
  8. Then repeat for columns 1, 2 and 3 obtaining, respectively, 484, 363, 242 and 121.
  9. Finally repeat for columns 7, 8 and 9 obtaining, respectively, 492, 369, 246 and 123.
  10. Fill the 4th & 8th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (7,6) &(8,9) and enter into Square A3.
  11. Fill the 3rd & 9th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (11,10),(13,12) & (14,15),(21,22) and enter into Square A4.
  12. Fill the 2nd & 10th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (24,23),(26,25) & (28,27) & (29,30),(36,37),(38,37) and enter into Square A5.
  13. And finally fill the 1st & 11th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (41,40),(43,42),(45,44),(52,51) & (53,54),(55,56),(57,58),(59,60) and enter into Square A6.
  14. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  15. Picture of squares
    Figure A
  16. Square A6 shows the 4 border squares in "border format".
  17. The complement table below also shows how the color pairs are layed out (for comparison with Square A6).
A1
 
 
 
 
  106 1 48
31 61 91
46 121 16
 
 
 
 
A2
102 5 72 492
  103 4 73 369
  104 3 74 246
  105 2 75 123
  106 1 76
35 3433 32 31 61 91 908988 87
46 121 16
  47 120 17 121
  48 119 18 242
  49 118 19 363
50 117 20484
484363242 121 123 246369492
A3
102 5 72
103 4 73
104 3 74
1057 2 116 75
8 106 1 76 114
35 3433 32 31 61 91 9089 8887
11346 121 16 9
47 115 120 6 17
48 119 18
49 118 19
50 117 20
A4
102 5 72
103 4 73
104 11 13 3110 112 74
14 1057 2 116 75 108
218 106 1 76 114 101
35 3433 32 31 61 91 9089 8887
10011346 121 16 9 22
10747 115 120 6 17 15
48 111 109 119 1210 18
49 118 19
50 117 20
A5
102 5 72
103 24 2628 4 95 97 99 73
29 104 1113 3110 112 74 93
3614 1057 2 116 75 108 86
3821 8 106 1 76 114 10184
35 3433 32 31 61 91 9089 8887
83100113 46 121 16 9 2239
85107 47 115 120 6 17 15 37
92 48 111 109 119 1210 18 30
49 98 9694 118 2725 23 19
50 117 20
A6
10241 4345 525 7178 808272
53103 24 2628 4 95 97 99 73 69
5529 104 1113 3110 112 74 9367
5736 14 1057 2 116 75 108 8665
593821 8 106 1 76 114 101 8463
35 3433 32 31 61 91 9089 8887
6283 100113 46 121 16 9 223960
6485107 47 115 120 6 17 15 3758
6692 48 111 109 119 1210 18 3056
6849 98 9694 118 27 25 23 19 54
5081 7977 70117 5144 4240 20
A7
10241 43 4552 5 71 78 80 82 72
53103 24 2628 4 95 97 99 73 69
5529 104 1113 3110 112 74 93 67
5736 14 1057 2 116 75 108 86 65
5938 218 106 1 76 114 10184 63
3534 33 32 31 61 91 908988 87
6283 100113 46 121 16 92239 60
64 85 10747 115 120 6 17 15 37 58
6692 48 111 109 119 1210 18 30 56
6849 98 9694 118 27 25 23 1954
50 81 79 77 70117 51 4442 40 20
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 4142 43 4445
 
101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 8180 79 7877
46 4748 49 5051 5253 54 5556 57 5859 60
61
76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part A7 of a 11x11 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part A8.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com