NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part M6

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, for n = 11, we can choose the complementary numbers (56,57,58,59,60,61,62,63,64,65,66) from the complimentary table below:

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

for the central column and not for the diagonal as was done for the regular wheel algorithm.. 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 (51,52,53,54,55) used to generate the left diagonal, 61 − 51.

In addition, 4n + 1 number behave differently from 4n + 3 in that in the former at least one number in the set must be less than or equal to 0, while in the latter all numbers from 1 to ½(n2 − 1) are usable. Figure A shows the connectivity of a 11x11 set. However, when Δ is an an odd number as for a 11x11 square at least one number in the set must be less than or equal to 0. This is shown in for a 5x5 square Part M1 and a variant square 5x5 Part M2.

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

A 11x11 Transposed Magic Square Using the Diagonals {46,47,48,49,50,61,72,73,74,75,76} and {55,54,53,52,51,61,71,70,69,68,67}

  1. Generate a 3x3 square using Δ=10, b=61 and a=62. (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 A2.
  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, 3 and 4 obtaining, respectively, 484, 363, 242 and 121.
  9. Finally repeat for columns 11, 10, 9 and 8 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, (45,44) &(42,43) 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, (35,34),(33,32) & (30,31),(28,29) 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),(25,24)(23,22) & (20,21),(18,19),(16,17) 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, (15,14),(13,12),(11,10),(9,8) & (6,7),(4,5),(2,3),(0,1) 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
 
 
 
 
  50 62 71
82 61 40
51 60 72
 
 
 
 
A2
46 66 67 492
  47 65 68 369
  48 64 69 246
  49 63 70 123
  50 62 71
86 8584 83 82 61 40 393837 36
51 60 72
  52 59 73 121
  53 58 74 242
  54 57 75 363
55 56 76484
484363242 121 123 246369492
A3
46 66 67
  47 65 68
  48 64 69
4945 63 78 70
42 50 62 71 80
86 8584 83 82 61 40 393837 36
7951 60 72 43
52 77 59 44 73
  53 58 74
  54 57 75
55 56 76
A4
46 66 67
  47 65 68
48 35 336490 88 69
30 4945 63 78 70 92
2842 50 62 71 80 94
86 8584 83 82 61 40 393837 36
937951 60 72 43 29
9152 77 59 44 73 31
53 87 89 58 3234 74
  54 57 75
55 56 76
A5
46 66 67
47 27 2523 65 100 98 96 68
20 48 3533 6490 88 69 102
1830 4945 63 78 70 92 104
1628 42 50 62 71 80 94106
86 8584 83 82 61 40 393837 36
1059379 51 60 72 43 2917
10391 52 77 59 44 73 31 19
101 53 87 89 58 3234 74 21
54 95 9799 57 2224 26 75
55 56 76
A6
4615 1311 966 114112 11010867
647 27 2523 65 100 98 96 68 116
420 48 3533 6490 88 69 102118
218 30 4945 63 78 70 92 104120
01628 42 50 62 71 80 94106 122
86 8584 83 82 61 40 393837 36
12110593 79 51 60 72 43 29171
11910391 52 77 59 44 73 31 193
117101 53 87 89 58 3234 74 215
11554 95 9799 57 22 24 26 75 7
55107 109111 11356 810 1214 76
A7
4615 13 119 66 114 112 110 108 67
647 27 2523 65 100 98 96 68 116
420 48 3533 6490 88 69 102 118
218 30 4945 63 78 70 92 104 120
016 2842 50 62 71 80 94106 122
8685 84 83 82 61 40 393837 36
121105 9379 51 60 72 432917 1
119 103 9152 77 59 44 73 31 19 3
117101 53 87 89 58 3234 74 21 5
11554 95 9799 57 22 24 26 757
55 107 109 111 11356 8 1012 14 76
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
 
122 121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103
20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40 4142 43 4445
 
102 101 100 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.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com