NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part P4

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). The only numbers retained are (59,61,63) with the remaining four complements (60,62), (58,64), (57,65) and (56,66) being replaced by another complementary pair 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, both 4n + 1 and 4n + 3 squares may be filled with the entire complement set. Previously the 4n + 3 square had to be filled with at least one 0 or negative number. See the 11x11 square.

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 {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 59 → 61 → 63 starting from the 5th row (Square A1).
  2. With 51, 52 and their complements generate a 3x3 square using Δ=7, b=61 and a=63 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 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 in this case is equal to 118 + 123 + 101 + 135. 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.
  6. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 236.
  7. Repeat for row 4 except subtract the value from 305 (the magic sum for a 5x5 internal square). This gives a value of 118.
  8. Do the same for rows 11, 10, 9 and 8 obtaining, respectively, 499, 373, 252 and 126.
  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, (34,38) &(36,37) 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, (31,35),(26,30) & (32,33),(28,29) 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, (23,27),(18,22)(25,24) & (20,21),(16,17),(12,13) 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, (15,19),(11,10),(62,83),(14,1) & (8,9),(6,7),(4,5),(2,3) 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 4 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
 
 
 
 
  52 63 68
77 61 45
54 59 70
 
 
 
 
A2
48 82 64 477
  49 76 65 359
  50 75 66 236
  51 69 67 118
  52 63 68
81 8079 78 77 61 45 444342 41
54 59 70
  55 53 71 126
  56 47 72 252
  57 46 73 374
58 40 74499
484363242 121 123 246369492
A3
48 82 64
  49 76 65
  50 75 66
5134 69 84 67
36 52 63 68 86
81 8079 78 77 61 45 444342 41
8554 59 70 37
55 88 53 38 71
  56 47 72
  57 46 73
58 40 74
A4
48 82 64
  49 76 65
50 31 267592 87 66
32 5134 69 84 67 90
2836 52 63 68 86 94
81 8079 78 77 61 45 444342 41
938554 59 70 37 29
8955 88 53 38 71 33
56 91 96 47 3035 72
  57 46 73
58 40 74
A5
48 82 64
49 23 1825 76 98 100 95 65
20 50 3126 7592 87 66 102
1632 5134 69 84 67 90 106
1228 36 52 63 68 86 94110
81 8079 78 77 61 45 444342 41
1099385 54 59 70 37 2913
10589 55 88 53 38 71 33 17
101 56 91 96 47 3035 72 21
57 99 10497 46 2422 27 73
58 40 74
A6
4815 1162 1482 12139 11210364
849 23 1825 76 98 10095 65114
620 50 3126 7592 87 66 102116
416 32 5134 69 84 67 90 106118
21228 36 52 63 68 86 94110 120
81 8079 78 77 61 45 444342 41
11910993 85 54 59 70 37 29133
11710589 55 88 53 38 71 33 175
115101 56 91 96 47 3035 72 217
11357 99 10497 46 24 2227 73 9
58107 11160 10840 183 1019 74
A7
4815 11 6214 82 121 39 112 103 64
849 23 1825 76 98 100 95 65 114
620 50 3126 7592 87 66 102 116
416 32 5134 69 84 67 90 106 118
212 2836 52 63 68 86 94110 120
8180 79 78 77 61 45 444342 41
119109 9385 54 59 70 372913 3
117 105 8955 88 53 38 71 33 17 5
115101 56 91 96 47 3035 72 21 7
11357 99 10497 46 24 22 27 739
58 107 111 60 10840 1 8310 19 74
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
 
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 P4 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com