NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part N10

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 (55,56,57,58,59) used to generate the left diagonal, 61 − 55 = 6.

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 {50,51,52,53,54,61,68,69,70,71,72} and {59,58,57,56,55,61,67,66,65,64,63}

  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=6, 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 354 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 231 which equals 123 + 108.
  7. Repeat for row 4 except subtract the value from 305 (the magic sum for a 5x5 internal square). This gives a value of 113.
  8. Do the same for rows 11, 10, 9 and 8 obtaining, respectively, 499, 378, 257 and 131.
  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, (31,40) &(38,39) 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, (37,36),(16,30) & (34,35),(32,33) 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, (28,29),(25,26)(17,27) & (23,24),(21,22),(19,20) 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, (13,14),(11,12),(8,9),(10,18) & (6,7),(4,5),(2,3),(0,1) and enter into Square A6.
  15. In addition, (15,107) 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
 
 
 
 
  54 62 67
74 61 48
55 60 68
 
 
 
 
A2
50 81 63 477
  51 80 64 354
  52 79 65 231
  53 73 66 113
  54 62 67
78 7776 75 74 61 48 474645 44
55 60 68
  56 49 69 131
  57 43 70 257
  58 42 71 378
59 41 72499
484363242 121 123 246369492
A3
50 81 63
  51 80 64
  52 79 65
  5331 73 82 66
  38 54 62 67 84
78 7776 75 74 61 48 474645 44
8355 60 68 39
  56 91 49 40 69
  57 43 70
  58 42 71
59 41 72
A4
50 81 63
  51 80 64
52 37 167992 86 65
34 5331 73 82 66 88
3238 4 62 67 84 90
78 7776 75 74 61 48 474645 44
898355 60 68 39 33
8756 91 49 40 69 35
57 85 106 43 3036 70
  58 42 71
59 41 72
A5
50 81 63
51 28 2517 80 95 9693 64
23 52 3716 7992 86 65 99
2134 5331 73 82 66 88 101
1932 38 54 62 67 84 90103
78 7776 75 74 61 48 474645 44
1028983 55 60 68 39 3320
10087 56 91 49 40 69 35 22
98 57 85 106 43 3036 70 24
58 94 97105 42 2726 29 71
59 41 72
A6
5013 118 1081 104113 110108 63
651 28 2517 80 95 9693 64116
423 52 3716 7992 86 65 99118
221 34 5331 73 82 66 88 101120
01932 38 54 62 67 84 90103 122
78 7776 75 74 61 48 474645 44
12110289 83 55 60 68 39 33201
11910087 56 91 49 40 69 35 223
11798 57 85 106 43 3036 70 245
11558 94 97105 42 27 2629 71 7
59109 111114 11241 189 1214 72
A7
5013 11 810 81 104 113 110 108 63
651 28 2517 80 95 96 93 64 116
423 52 3716 7992 86 65 99 118
221 34 5331 73 82 66 88 101 120
019 3238 54 62 67 84 90103 122
7877 76 75 74 61 48 474645 44
121102 8983 55 60 68 393320 1
119 100 8756 91 49 40 69 35 22 3
11798 57 85 106 43 3036 70 24 5
11558 94 97105 42 27 26 29 717
59 109 111 114 11241 18 912 14 72
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
 
102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82
4142 43 44 45 46 4748 49 5051 5253 54 5556 57 5859 60
61
8180 79 78 77 76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part N9 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com