NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part N9

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 (54,55,56,57,58) used to generate the left diagonal, 61 − 54 = 7.

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 {49,50,51,52,53,61,69,70,71,72,73} and {58,57,56,55,54,61,68,67,66,65,64}

  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=7, 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 477 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 + 114). 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, 499, 373, 252 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, (40,39) &(37,38) 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, (35,34),(27,36) & (32,33),(30,31) 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, (29,28),(25,24)(27,26) & (22,23),(20,21),(18,19) 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, (8,16),(14,15),(12,13),(10,11) & (6,7),(4,5),(2,3),(0,1) and enter into Square A6.
  15. In addition, (9,113) 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
 
 
 
 
  53 62 68
76 61 46
54 60 69
 
 
 
 
A2
49 81 64 477
  50 75 65 359
  51 74 66 236
  52 63 67 123
  53 62 68
80 7978 77 76 61 46 454443 42
54 60 69
  55 59 70 121
  56 48 71 252
  57 47 72 373
58 41 73499
484363242 121 123 246369492
A3
49 81 64
  50 75 65
  51 74 66
  5240 63 83 67
  37 53 62 68 85
80 7978 77 76 61 46 454443 42
8454 60 69 38
  55 82 59 39 70
  56 48 71
  57 47 72
58 41 73
A4
49 81 64
  50 75 65
51 35 277486 88 66
32 5240 63 83 67 90
3037 53 62 68 85 92
80 7978 77 76 61 46 454443 42
918454 60 69 38 31
8955 82 59 39 70 33
56 87 95 48 3634 71
  57 47 72
58 41 73
A5
49 81 64
50 29 2517 75 96 98 94 65
22 51 3527 7486 88 66 100
2032 5240 63 83 67 90 102
1830 37 53 62 68 85 92104
80 7978 77 76 61 46 454443 42
1039184 54 60 69 38 3119
10189 55 82 59 39 70 33 21
99 56 87 95 48 3634 71 23
57 93 97105 47 2624 28 72
58 41 73
A6
498 1412 1081 111109 107106 64
650 29 2517 75 96 9894 65116
422 51 3527 7486 88 66 100118
220 32 5240 63 83 67 90 102120
01830 37 53 62 68 85 92104 122
80 7978 77 76 61 46 454443 42
12110391 84 54 60 69 38 31191
11910189 55 82 59 39 70 33 213
11799 56 87 95 48 3634 71 235
11557 93 97105 47 26 2428 72 7
58114 108110 11241 1113 1516 73
A7
498 14 1210 81 111 109 107 106 64
650 29 2517 75 96 98 94 65 116
422 51 3527 7486 88 66 100 118
220 32 5240 63 83 67 90 102 120
018 3037 53 62 68 85 92104 122
8079 78 77 76 61 46 454443 42
121103 9184 54 60 69 383119 1
119 101 8955 82 59 39 70 33 21 3
11799 56 87 95 48 3634 71 23 5
11557 93 97105 47 26 24 28 727
58 114 108 110 11241 11 1315 16 73
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 N9 of a 11x11 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part N10.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com