NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part N8

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, the spoke of the central column is no longer the adjacent numbers (56,57,58,59,60,61,62,63,64,65,66). At least one pair of complements must be retained in the central column which may be (57,58,59,60,61,62,63,64,65) for the replacement of one, (58,59,60,61,62,63,64) for the replacement of two, (59,60,61,62,63) for the replacement of three and (60,61,62) for the replacement of sets of complementary pairs from the list:

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 (53,54,55,56,57) used to generate the left diagonal, 61 − 53 = 8.

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.

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 {48,49,50,51,52,61,70,71,72,73,74} and {57,56,55,54,53,61,69,68,67,66,65}

  1. To the center column of the internal 3x3 square fill numbers ½(n2-1) to ½(n2+3) in consecutive order starting at the bottom middle cell of the 3x3 internal square and proceeding to the top middle cell of the 3x3 internal square using the numbers listed in the complementary table described above, as for example using n = 11. For a 11x11 square the numbers in the center column correspond to 60 → 61 → 62 starting from the 5th row (Square A1).
  2. With 52, 53 and their complements generate a 3x3 square using Δ=8, b=61 and a=62 so that the sum of each column, row and diagonal of the 3x3 square sums up to 183, the sum of the internal 3x3 square within a 11x11 square (Square A1).
  3. 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.
  4. 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 482 which although not divisible by 4 requires three numbers that are equal and a fourth nonequal number to fill up that line, which in for example might be (121 x 3 + 119). See Figure A2.
  5. Repeat for row 2 except subtract the value from 549 (the magic sum for a 9x9 internal square). This gives a value of 359 which equals 123 x 2 + 113.
  6. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 123 x 2.
  7. Repeat for row 4 except subtract the value from 305 (the magic sum for a 5x5 internal square). This gives a value of 123.
  8. Do the same for rows 11, 10, 9 and 8 obtaining, respectively, 494, 363, 242 and 121.
  9. Then repeat for columns 1, 2, 3 and 4 obtaining, respectively, 484, 363, 242 and 121.
  10. Finally repeat for columns 11, 10, 9 and 8 obtaining, respectively, 492, 369, 246 and 123.
  11. Fill the 4th & 8th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (38,37) &(33,34) and enter into Square A3.
  12. Fill the 3rd & 9th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (30,29),(32,31) & (27,28),(24,25) and enter into Square A4.
  13. Fill the 2nd & 10th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (23,22),(21,20)(26,35) & (18,19),(16,17),(14,15) and enter into Square A5.
  14. And finally fill the 1st & 11th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (12,13),(10,11),(8,9),(36,39) & (6,7),(4,5),(2,3),(0,1) and enter into Square A6.
  15. In addition, (45,77) cannot be included into the magicsquare since n is of type 4n + 3.
  16. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  17. Picture of squares
    Figure A
  18. Square A6 shows the 4 border squares in "border format".
  19. The complement table below also shows how the color pairs are layed out (for comparison with Square A6).
A1
 
 
 
 
  52 62 69
78 61 44
53 60 70
 
 
 
 
A2
48 76 65 482
  49 75 66 359
  50 64 67 246
  51 63 68 123
  52 62 69
82 8180 79 78 61 44 434241 40
53 60 70
  54 59 71 121
  55 58 72 242
  56 47 73 373
57 46 74494
484363242 121 123 246369492
A3
48 76 65
  49 75 66
  50 64 67
  5138 63 85 68
  33 52 62 69 89
82 8180 79 78 61 44 434241 40
88 53 60 70 34
  54 84 59 37 71
  55 58 72
  56 47 73
57 46 74
A4
48 76 65
  49 75 66
50 30 326491 93 67
27 5138 63 85 68 95
2433 52 62 69 89 98
82 8180 79 78 61 44 434241 40
978853 60 70 34 25
9454 84 59 37 71 28
55 92 90 58 3129 72
  56 47 73
57 46 74
A5
48 76 65
49 23 2126 75 87 102 100 66
18 50 3032 6491 93 67 104
1627 5138 63 85 68 95 106
1424 33 52 62 69 89 98108
82 8180 79 78 61 44 434241 40
1079788 53 60 70 34 2515
10594 54 84 59 37 71 28 17
103 55 92 90 58 3129 72 19
56 99 10196 47 3520 22 73
57 46 74
A6
4812 108 3676 83113 11110965
649 23 2126 75 87 102100 66116
418 50 3032 6491 93 67 104118
216 27 5138 63 85 68 95 106120
01424 33 52 62 69 89 98108 122
82 8180 79 78 61 44 434241 40
12110797 88 53 60 70 34 25151
11910594 54 84 59 37 71 28 173
117103 55 92 90 58 3129 72 195
11556 99 10196 47 35 2022 73 7
57110 112114 8646 399 1113 74
A7
4812 10 836 76 83 113 111 109 65
649 23 2126 75 87 102 100 66 116
418 50 3032 6491 93 67 104 118
216 27 5138 63 85 68 95 106 120
014 2433 52 62 69 89 98108 122
8281 80 79 78 61 44 434241 40
121107 9788 53 60 70 342515 1
119 105 9454 84 59 37 71 28 17 3
117103 55 92 90 58 3129 72 19 5
11556 99 10196 47 35 20 22 737
57 110 112 114 8646 39 911 13 74
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 N8 of a 11x11 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part N9.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com