NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part R5

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), i.e., ½(n2-n+2) to ½ n2+n, but may be chosen from any other consecutive group of numbers, which in our case may be (49,50,51,52,53,61,69,70,71,72,73)(this page) or (50,51,52,53,54,61,68,69,70,71,72) (next page).

Each will be treated separately since 4n + 3, where in this case n = 2 may be filled with every number in the complementary set 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

or with one number being either 0 or 1 and its complement. In the 4n + 1 numbers, however, both squares use all the numbers in their complementary set.

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.

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 {43,44,45,46,47,61,75,76,77,78,79} 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 55 → 61 → 67 starting from the 5th row (Square A1).
  2. With 55, 67 and their complements generate a 3x3 square using Δ=6, b=61 and a=69 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. Then begin filling up the square by adding up the entries on the first row and subtracting from 671 (the magic sum for a 11x11 square). This affords the value 492 for example equals 123 x 4. 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 369.
  6. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 246.
  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, 484, 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, (34,33) &(29,30) 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, (28,27),(26,25) & (17,18),(15,16) 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, (24,23),(22,21),(20,19) & (13,14),(11,12),(9,10) 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, (60,54),(80,74),(86,90),(87,91) & (1,2),(3,4),(5,6),(7,8) and enter into Square A6.
  15. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  16. Picture of squares
    Figure A
  17. Square A6 shows the 5 border squares in "border format".
  18. The complement table below also shows how the color pairs are layed out (for comparison with Square A6).
A1
 
 
 
 
  47 69 67
81 61 41
55 53 75
 
 
 
 
A2
43 73 63 492
  44 72 64 369
  45 71 65 246
  46 70 66 123
  47 69 67
85 8483 82 81 61 41 403938 37
55 53 75
  56 52 76 121
  57 51 77 242
  58 50 78 363
59 49 79484
484363242 121 123 246369492
A3
43 73 63
  44 72 64
  45 71 65
4634 70 89 66
29 47 69 67 93
85 8483 82 81 61 41 403938 37
9255 53 75 30
56 88 52 33 76
  57 51 77
  58 50 78
59 49 79
A4
43 73 63
  44 72 64
45 28 267197 95 65
17 4634 70 89 66 105
1529 47 69 67 93 107
85 8483 82 81 61 41 403938 37
1069255 53 75 30 16
10456 88 52 33 76 18
57 94 96 51 2527 77
  58 50 78
59 49 79
A5
43 73 63
44 24 2220 72 103 101 99 64
13 45 2826 7197 95 65 109
1117 4634 70 89 66 105 111
915 29 47 69 67 93 107113
85 8483 82 81 61 41 403938 37
11210692 55 53 75 30 1610
110104 56 88 52 33 76 18 12
108 57 94 96 51 2527 77 14
58 98 100102 50 1921 23 78
59 49 79
A6
4360 8086 8773 3132 486863
144 24 2220 72 103 10199 64121
313 45 2826 7197 95 65 109119
511 17 4634 70 89 66 105 111117
7915 29 47 69 67 93 107113 115
85 8483 82 81 61 41 403938 37
114112106 92 55 53 75 30 16108
116110104 56 88 52 33 76 18 126
118108 57 94 96 51 2527 77 144
12058 98 100102 50 19 2123 78 2
5962 4236 3549 9190 7454 79
A7
4360 80 8687 73 31 32 48 68 63
144 24 2220 72 103 101 99 64 121
313 45 2826 7197 95 65 109 119
511 17 4634 70 89 66 105 111 117
79 1529 47 69 67 93 107113 115
8584 83 82 81 61 41 403938 37
114112 10692 55 53 75 301610 8
116 110 10456 88 52 33 76 18 12 6
118108 57 94 96 51 2527 77 14 4
12058 98 100102 50 19 21 23 782
59 62 42 36 3549 91 9074 54 79
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
 
121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104
19 20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40 41
 
103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81
42 43 44 4546 4748 49 5051 5253 54 5556 57 5859 60
61
8079 7877 76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part R5 of a 11x11 Magic Square Wheel Spoke Shift method. To go to Part R6 of an 11x11 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com