NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Picture of a wheel

How to Spoke Shift 9x9 Magic Squares-(Part E)

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 two new methods used for the construction of wheel type squares except that the initial spoke parts are added in a somewhat different manner than in the original wheel method. The first method consists of pairing numbers in complementary fashion, partitioning these complementary pairs into groups, generating in the spoke and then filling in the non spoke cells with the remaining complementary pairs as was done in the original method. The difference between this type of square and the original is that numbers less than or equal to 0 and their complements are can be part of the square. Since the number of cells in an nxn magic squares is n then a complementary pair containing 0 and/or negative numbers are required for generating these type of magic squares.

The second method consists of transposing rows and columns around to generate a magic square where the spoke numbers have been inverted. Method one generates border squares where the internal squares and the external squares are magic. Method 2 produces only one magic square, the external one. The internal squares are all non magic.

In addition, the diagonal pairs are obtained from the complementary table using what I call a "Cross-Over" method shown below. For a square with n = 9, there are 25 sets of pairs. These pairs and their complements make up entries to the diagonal cells. A diagram of the {23,21,19,17} and {12,14,16,18} connectivity is shown below in Figure A.

The new magic squares with n = 9 are constructed as follows using a complimentary table as a guide (where 21 follows 20 on the third line).


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62
 
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42
  1. The center column is filled with the group of numbers ½ (n2-n+2) to ½(n2+n) in consecutive order starting at the bottom cell and proceeding to the top cell from the numbers listed in the complementary table described above, for example using n = 9. For a 9x9 square the numbers in the center column correspond to 37 → 38 → 39 → 40 → 41 → 42 → 43 → 44 → 45 starting from the bottom (Square A1).
  2. 24 pairs are left with which to construct the spoke and fill in the non-spoke cells. Table Fa tells us that for n = 9 there are 25 sets that can generate a "Cross-Over" of evenly spaced numbers. The spoke cells are chosen from a group of 25 pairs of evenly spaced numbers. In this exercise we pick the 22nd pair (23 → 21 → 19 → 17) and (12 → 14 → 16 → 18) where the crossover point between (28,29) is the "Cross-Over". The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A2. The numbers 23, 21, 19 and 17 are added, in that order, down to the right and 12, 14, 16 and 18 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {-6,-5,-4,-3} to the center row with 0 to the right of 41, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {85,86,87,88} to the left of 41 (Square A3).
  4. To fill up the rest of the square work with the internal square first, i.e., 5x5 where (15 is paired with 20) and (25 with 26) along with their complements in the same row or column to form Square A4. Note that {25,26} are adjacent on the complementary table while {15,20} are 5 units away.
  5. Fill in the next internal square 7x7 by pairing {1 with 6}, {2 with 7}, {26 with 27}, and {28 with 29}.
  6. Fill in the external square 9x9 by pairing {3 with 8}, {4 with 9}, {5 with 10}, {30 with 31}, {32 with 33},and {34 with 35}. Not all the complementary pairs are used for this square,i.e., 11,13,22 and 26 are unincluded. The picture below shows the physical connectivity.
  7. The portion of the complementary table (just the top set of numbers since the same applies to the bottom set) showing the connectivity of the non-spoke numbers and the "Cross-Over" is shown as a little red cross, is summarized as:

    Picture of a wheel
  8. Note Note that the middle row from -3 to -6 are not included.
  9. The result of these operations is a wheel with a shifted spoke where the numbers in the diagonal of the regular wheel 37 → 38 → 39 → 40 → 41 → 42 → 43 → 44 → 45 have been transposed or shifted to a column.
  10. The square that is produced via this method is a border square, since the 3x3 square has an S = 123, the 5x5 has an S = 205, the 7x7 has an S = 287 and the 9x9 has an S = 369. These border squares are shown in Square A6.
A1
45
44
43
42
41
40
39
38
37
A2
23 45 70
21 44 68
19 43 66
17 42 64
41
18 40 65
16 39 63
14 38 61
12 37 59
A3
23 45 70
21 44 68
19 43 66
17 42 64
8586 87 88 41 -6 -5-4 -3
18 40 65
16 39 63
14 38 61
12 37 59
A4
23 45 70
21 44 68
1915 43 62 66
25 17 42 64 57
8586 87 88 41 -6 -5-4 -3
5818 40 65 24
16 67 39 20 63
14 38 61
12 37 59
A5
23 45 70
21 12 44 75 76 68
271915 43 62 66 55
2925 17 42 64 57 53
8586 87 88 41 -6 -5-4 -3
545818 40 65 2428
5616 67 39 20 63 26
14 81 80 38 76 61
12 37 59
A6
23 3 45 45 7273 74 70
31 21 1 244 7576 6851
3327 1915 43 62 66 55 49
3529 25 17 42 64 57 5347
8586 87 88 41 -6 -5-4 -3
485458 18 40 65 24 2834
5056 16 67 39 20 63 2632
5214 81 80 38 76 61 30
12 79 7877 37 109 8 59
A6 Border
23 3 45 45 7273 74 70
31 21 12 44 75 76 68 51
3327 1915 43 62 66 55 49
3529 25 17 42 64 57 5347
8586 87 88 41 -6 -5-4-3
4854 5818 40 65 242834
5056 16 67 39 20 63 2632
5214 81 80 38 76 61 30
12 79 7877 37 10 9 8 59
-6 -5 -4 -3 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
 
88 87 86 85 ... 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61
 
22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

Conversion of the 9x9 into its transposed opposite

Generation of a 9x9 transposed opposite can also follow the route used above. Unfortunately as n > 5 their generation becomes more and more complicated. A method that obviates this is to transpose columns followed by rows. This generates a new square which is not a border square. Only the external square is magic.

  1. Take square A6 and transpose (column 1 with column 4), (column 2 with column 3), (column 6 with column 9) and (column 7 with column 8) to get Square A7.
  2. Take square A7 and transpose (row 1 with row 4), (row 2 with row 3), (row 6 with row 9) and (row 7 with row 8) to get Square A8.
  3. In a sense A6 has been imploded or everted into A8, i.e., A6 and A8 below are opposites.
A6
23 3 45 45 7273 74 70
31 21 1 244 7576 6851
3327 1915 43 62 66 55 49
3529 25 17 42 64 57 5347
8586 87 88 41 -6 -5-4 -3
485458 18 40 65 24 2834
5056 16 67 39 20 63 2632
5214 81 80 38 76 61 30
12 79 7877 37 109 8 59
A7
5 4 323 45 7074 73 72
2 1 21 31 44 5168 76 75
1519 2733 43 49 55 66 62
1725 29 35 42 47 53 5764
8887 86 85 41 -3 -4 -5-6
185854 48 40 34 28 2465
6716 56 50 39 32 26 6320
8081 14 52 38 3061 6 7
77 78 7912 37 59 89 10
A8
1725 29 35 42 47 53 5764
1519 2733 43 49 55 66 62
2 1 21 31 44 5168 76 75
5 4 323 45 7074 73 72
8887 86 85 41 -3 -4 -5-6
77 78 7912 37 59 89 10
8081 14 52 38 3061 6 7
6716 56 50 39 32 26 6320
185854 48 40 34 28 2465

The result is a new square conforming to the same complementary table above which obviates the need to go thru the complicared rigmarole of filling in the non-spoke cells which appears to be more difficult to do.

This completes Part E of a 9x9 Magic Square Wheel Spoke Shift method. To go back to 9x9 Part D.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com