NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part R7

Picture of a wheel

How to Spoke Shift 13x13 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 (79,80,81,82,83,84,85,86,87,88,89,90,91), 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 (72,73,74,75,76,77,85,93,94,95,96,97,98)(this page) or (71,72,73,74,75,76,85,94,95,96,97,98,99) (next page).

Each will be treated separately since 4n + 1, where in this case n = 3 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

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 (78,79,80,81,82,83) used to generate the left diagonal, 85 − 78 = 7.

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

{65,66,67,68,69,70,85,100,101,102,103,104,105} and {83,82,81,80,79,85,92,91,90,89,88,87}

  1. To the internal 3x3 square fill the numbers ½(n2-1) to ½(n2+3) in consecutive order using the numbers listed in the complementary table described above, n = 13. For a 13x13 square the numbers in the center column correspond to 78 → 85 → 92 (Square A1). Initially only this 3x3 square is used for the entire 13x13 square.
  2. With 78, 92 and their complements generate a 3x3 square using Δ=7, b=85 and a=93 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 13x13 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 1105 (the magic sum for a 13x13 square). This affords the value 855 for example equals 171 x 5 . See Figure A2.
  5. Repeat for row 2 except subtract the value from 935 (the magic sum for a 11x11 internal square). This gives a value of 684.
  6. Repeat for row 3 except subtract the value from 765 (the magic sum for a 9x9 internal square). This gives a value of 513.
  7. Repeat for row 4 except subtract the value from 595 (the magic sum for a 7x7 internal square). This gives a value of 342.
  8. Repeat for row 5 except subtract the value from 425 (the magic sum for a 5x5 internal square). This gives a value of 171.
  9. Do the same for rows 13, 11,10, 9 and 8 obtaining, respectively, 845, 676, 507, 338 and 169.
  10. Then repeat for columns 1, 2, 3 and 4 obtaining, respectively, 845, 676, 507, 338 and 169.
  11. Finally repeat for columns 11, 10, 9 and 8 obtaining, respectively, 855, 684, 513, 342 and 171.
  12. Fill the 5th & 9th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (55,54) & (52,53) and enter into Square A3.
  13. Fill the 4th & 10th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (84,71),(1,12) &(57,64),(56,51) and enter into Square A3. These numbers are listed in bold since they are non-adjacent numbers in the complementary table and are numbers required to give the desired sums.
  14. Fill the 3rd & 11th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (50,49),(48,47),(46,45) & (43,44),(41,42),(39,40) and enter into Square A3.
  15. Fill the 2nd & 12th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (38,37),(36,35),(34,33),(32,31) & (29,30),(27,28),(25,26),(23,24) and enter into Square A3.
  16. And finally fill the 1st & 13th rows & columns with the pairs/complements from the complement list corresponding to the requisite sums, (22,21),(20,19),(18,17),(16,15),(14,13) & (10,11),(8,9),(6,7),(4,5),(2,3) and enter into Square A3.
  17. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers. The same for their complements.
  18. Picture of squares
    Figure A
  19. Square A4 shows the 6 border squares in "border format".
  20. The complement table below also shows how the color pairs are layed out (for comparison with Square A3).
A1
70 93 92
107 85 63
78 77 100
A2
65 98 87 855
66 97 88 684
  67 96 89 513
  68 95 90 342
  69 94 91 171
  70 93 92
112111 110109 108 107 85 63 626160 5958
78 77 100
  79 76 101 169
  80 75 102 338
  81 74 103 507
82 73 104 676
83 72 105 845
845676507 338169 171 342513684 855
A3
6522 20 18 1614 98 157155 153 151 149 87
10 66 38 36 3432 97 139 137 135 133 88160
829 67 50 4846 96125123 121 89 141 162
627 43 68 841 95 158 99 90 127 143 164
425 41 57 6955 94 116 91 113 129 145 166
223 39 56 52 70 93 92 118 114 131 147 168
112111 110109 108 107 85 63 626160 5958
167 146 130 119 117 78 77 100 53 51 40 24 3
165144 128 106 79 115 76 54 10164 42 265
163142 126 8086 169 75 12 71 10244 28 7
161140 81120 122124 744547 49 103 30 9
159 82 132 134 136138 73 31 33 3537 10411
83148 150 152 154156 72 1315 17 19 21 105
A4 Border
6522 20 18 1614 98 157155 153151 149 87
10 66 38 36 3432 97 139 137 135 133 88160
829 67 50 4846 96125123 121 89 141 162
627 43 68 841 95 158 99 90 127 143 164
425 41 57 6955 94 116 91 113129 145 166
223 39 56 52 70 93 92 118 114 131 147 168
112111 110109 108 107 85 63 626160 5958
167 146 130 119 117 78 77 100 53 51 40 24 3
165144 128 106 79 115 76 54 10164 42 265
163142 126 80 86 169 75 12 71 10244 28 7
161140 81120 122124 744547 49 103 30 9
159 82 132 134 136138 73 31 33 3537 10411
83148 150 152 154156 72 1315 1719 21 105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
 
169 168 167 166 165 164 163 162 161 160 159 158 157 156 155 154 153 152
19 20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
 
151 150 149 148 147 146145 144143 142 141 140 139 138 137 136 135 134 133 132 131 130
4142 43 44 4546 4748 49 5051 5253 54 5556
 
129 128127 126125 124 123122 121 120119 118117 116 115114
57 5859 60 6162 63 6465 66 6768 69 7071 72 7374 75 7677 78 7980 81 8283 84
85
113 112111 110 109108 107 106105 104 103102 101 10099 98 9796 95 9493 92 9190 89 8887 86

This completes the Part R7 of a 13x13 Magic Square Wheel Spoke Shift method. To go to Part R8 of an 13x13 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com