NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part N7

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 (52,53,54,55,56) used to generate the left diagonal, 61 − 52 = 9.

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 {47,48,49,50,51,61,71,72,73,74,75} and {56,55,54,53,52,61,70,69,68,67,66}

  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 51, 52 and their complements generate a 3x3 square using Δ=9, 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 482 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 + 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 123 x 3.
  6. Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 123 x 2.
  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, 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, (45,44) &(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, (33,32),(31,30) & (28,29),(26,27) 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, (25,24),(23,22)(21,20) & (12,13),(10,11),(8,9) 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, (34,37),(18,19),(16,17),(14,15) & (0,1),(2,3),(4,5),(6,7) and enter into Square A6.
  15. In addition, (43,79) 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
 
 
 
 
  51 62 70
80 61 42
52 60 71
 
 
 
 
A2
47 76 66 482
  48 65 67 369
  49 64 68 246
  50 63 69 123
  51 62 70
84 8382 81 80 61 42 414039 38
52 60 71
  53 59 72 121
  54 58 73 242
  55 57 74 363
56 46 75494
484363242 121 123 246369492
A3
47 76 66
  48 65 67
  49 64 68
5045 63 78 69
35 51 62 70 87
84 8382 81 80 61 42 414039 38
8652 60 71 36
53 77 59 44 72
  54 58 73
  55 57 74
56 46 75
A4
47 76 66
  48 65 67
49 33 316492 90 68
28 5045 63 78 69 94
2635 51 62 70 87 96
84 8382 81 80 61 42 414039 38
958652 60 71 36 27
9353 77 59 44 72 29
54 89 91 58 3032 73
  55 57 74
56 46 75
A5
47 76 66
48 25 2321 65 102 100 98 67
12 49 3331 6492 90 68 110
1028 5045 63 78 69 94 112
826 35 51 62 70 87 96114
84 8382 81 80 61 42 414039 38
1139586 52 60 71 36 279
11193 53 77 59 44 72 29 11
109 54 89 91 58 3032 73 13
55 97 99101 57 2022 24 74
56 46 75
A6
4734 1816 1476 107105 1038566
048 25 2321 65 102 10098 67122
212 49 3331 6492 90 68 110120
410 28 5045 63 78 69 94 112118
6826 35 51 62 70 87 96114 116
84 8382 81 80 61 42 414039 38
11511395 86 52 60 71 36 2797
11711193 53 77 59 44 72 29 115
119109 54 89 91 58 3032 73 133
12155 97 99101 57 20 2224 74 1
5688 104106 10846 1517 1937 75
A7
4734 18 1614 76 107 105 103 85 66
048 25 2321 65 102 100 98 67 122
212 49 3331 6492 90 68 110 120
410 28 5045 63 78 69 94 112 118
68 2635 51 62 70 87 96114 116
8483 82 81 80 61 42 414039 38
115113 9586 52 60 71 36279 7
117 111 9353 77 59 44 72 29 11 5
119109 54 89 91 58 3032 73 13 3
12155 97 99101 57 20 22 24 741
56 88 104 106 10846 15 1719 37 75
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 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 N7 of a 11x11 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part N8.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com