NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part Q4

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. However, the spoke on the consisting of the central column is no longer the adjacent numbers (56,57,58,59,60,61,62,63,64,65,66), where the number 59 correponding to ½(n2-3) is retained in (59,61,63) the next number on the list is ½(n2-5), i.e., 58, so that now the central column is (..,58,59,61,63,64,..) and (60,62) no longer belongs on the "spoke" but are replaced by a pair of complementary numbers (the ..) (see Square A2 below). Furthermore, the rest of the numbers belonging in the square are generated from the list 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

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, 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 {47,48,49,50,51,61,71,72,73,74,75} 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 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 by adding up the entries on the first row and subtracting from 671 (the magic sum for a 11x11 square). This affords the value 482 for example equals 121x3 + 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.
  6. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 241.
  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, 373, 247 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) &(35,36) 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, (44,60),(14,1) & (33,34),(31,32) 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, (2,5),(6,9),(3,4) & (29,30),(27,28),(25,26) 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, (15,16),(11,12),(7,8),(10,13) & (17,18),(19,20),(21,22),(23,24) 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
 
 
 
 
  51 63 69
79 61 43
53 59 71
 
 
 
 
A2
47 77 65 482
  48 76 66 359
  49 70 67 241
  50 64 68 118
  51 63 69
83 8281 80 79 61 43 424140 39
53 59 71
  54 58 72 121
  55 52 73 247
  56 46 74 373
57 45 75494
484363242 121 123 246369492
A3
47 77 65
  48 76 66
  49 70 67
5038 64 85 68
35 51 63 69 87
83 8281 80 79 61 43 424140 39
8653 59 71 36
54 84 58 37 72
  55 52 73
  56 46 74
57 45 75
A4
47 77 65
  48 76 66
49 44 1470121 62 67
33 5038 64 85 68 89
3135 51 63 69 87 91
83 8281 80 79 61 43 424140 39
908653 59 71 36 32
8854 84 58 37 72 34
55 78 108 52 160 73
  56 46 74
57 45 75
A5
47 77 65
48 2 63 76 118 113 117 66
29 49 4414 70121 62 67 93
2733 5038 64 85 68 89 95
2531 35 51 63 69 87 9197
83 8281 80 79 61 43 424140 39
969086 53 59 71 36 3226
9488 54 84 58 37 72 34 28
92 55 78 108 52 160 73 30
56 120 116119 46 49 5 74
57 45 75
A6
4715 117 1077 109114 11010665
1748 2 63 76 118 113117 66105
1929 49 4414 70121 62 67 93103
2127 33 5038 64 85 68 89 95101
232531 35 51 63 69 87 9197 99
83 8281 80 79 61 43 424140 39
989690 86 53 59 71 36 322624
1009488 54 84 58 37 72 34 2822
10292 55 78 108 52 160 73 3020
10456 120 116119 46 4 95 74 18
57107 111115 11245 138 1216 75
A7
4715 11 710 77 109 114 110 106 65
1748 2 63 76 118 113 117 66 105
1929 49 4414 70121 62 67 93 103
2127 33 5038 64 85 68 89 95 101
2325 3135 51 63 69 87 9197 99
8382 81 80 79 61 43 424140 39
9896 9086 53 59 71 363226 24
100 94 8854 84 58 37 72 34 28 22
10292 55 78 108 52 160 73 30 20
10456 120 116119 46 4 9 5 7418
57 107 111 115 11245 13 812 16 75
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 4142 43
 
103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 8180 79
44 4546 4748 49 5051 5253 54 5556 57 5859 60
61
7877 76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part Q4 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com