NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

BORDER and SPOKE SHIFT Part A8

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 (55,54,53,52,51,61,71,70,69,68,67) 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 (55,54,53,52,51), i.e., Δ = 61 − 51 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 {107,108,109,110,111,61,11,12,13,14,15} and {55,54,53,52,51,61,71,70,69,68,67}

  1. Generate a 3x3 square using Δ=10, 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) & (9,10) 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, (16,8),(66,72) & (17,18),(19,20) 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, (27,26),(29,28),(31,30) & (40,41),(42,43),(44,45),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, (33,32),(35,34),(37,36),(39,38) & (46,47),(48,49),(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
 
 
 
 
  111 1 71
21 61 101
51 121 11
 
 
 
 
A2
107 5 67 492
  108 4 68 369
  109 3 69 246
  105 2 75 123
  110 1 70
25 2423 22 21 61 101 1009998 97
51 121 11
  52 120 12 121
  53 119 13 242
  54 118 14 363
55 117 15484
484363242 121 123 246369492
A3
107 5 67
108 4 68
109 3 69
1107 2 116 70
9 111 1 71 113
25 2423 22 21 61 101 10099 9897
11251 121 11 10
52 115 120 6 12
53 119 13
54 118 14
55 117 15
A4
107 5 67
108 4 68
109 16 66 350 114 69
17 1107 2 116 70 105
199 111 1 71 113 103
25 2423 22 21 61 101 10099 9897
10211251 121 11 10 20
10452 115 120 6 12 18
53 106 56 119 728 13
54 118 14
55 117 15
A5
107 5 67
108 27 2931 4 92 94 96 68
40 109 1666 350 114 69 82
4217 1107 2 116 70 105 80
4419 9 111 1 71 113 10378
25 2423 22 21 61 101 10099 9897
77102112 51 121 11 10 2045
79104 52 115 120 6 12 18 43
81 53 106 56 119 728 13 41
54 95 9391 118 3028 26 14
55 117 15
A6
10733 3537 395 8486 889067
46108 27 2931 4 92 94 96 68 76
4840 109 1666 350 114 69 8274
574217 1107 2 116 70 105 8065
594419 9 111 1 71 113 103 7863
25 2423 22 21 61 101 10099 9897
6277 102112 51 121 11 10 204560
6479104 52 115 120 6 12 18 4358
7381 53 106 56 119 728 13 4149
7554 95 9391 118 3028 26 14 47
5589 8785 83117 3836 3432 15
A7
10733 35 3739 5 84 86 88 90 67
46108 27 2931 4 92 94 96 68 76
4840 109 1666 350 114 69 82 74
5742 17 1107 2 116 70 105 80 65
5944 199 111 1 71 113 10378 63
2524 23 22 21 61 101 1009998 97
6277 102112 51 121 11 102045 60
64 79 10452 115 120 6 12 18 43 58
7381 53 106 56 119 728 13 41 49
7554 95 9391 118 30 28 26 1447
55 89 87 85 83117 38 3634 32 15
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 A8 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com