NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part D7

Picture of a wheel

How to Spoke Shift 9x9 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 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 may be included in the square.

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.

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 "Multi-Cross-Over" method shown below. For a square with n = 9, there are 29 sets of pairs which are shown in as evenly spaced pairs. These pairs and their complements make up entries to the diagonal cells. This site uses the {31,27,29,31} and {26,28,30,32} pairs along with the connectivity of all the numbers used for the squares (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

A 9x9 Magic Square Using the Pairs {31,27,29,31} and {26,28,30,32}

  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 Fg tells us that for n = 9 there are 29 sets that can generate a "Multi-Cross-Over or terminus" of evenly spaced numbers. The spoke cells are chosen from a group of 29 pairs of evenly spaced numbers. In this exercise we pick the 25th pair (25 → 27 → 29 → 31) and (26 → 28 → 30 → 32) where 5 Multi-Crossover points are present at (26,27), (27,28), (28,29), (29,30) and (30,31). The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A2. The numbers 9, 7, 5 and 3 are added, in that order, down to the right and 10, 8, 6 and 4 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {10,11,12,13} to the center row with 10 to the right of 41, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {69,70,71,72} 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 (35 is paired with 36) and (21 with 24) along with their complements in the same row or column to form Square A4. Note that {35,36} are adjacent on the complementary table while {21,24} are 4 units away.
  5. Fill in the next internal square 7x7 by pairing {33 with 34}, {7 with 8}, {20 with 23}, and {19 with 22}.
  6. Fill in the external square 9x9 by pairing {0 with 1}, {2 with 3}, {4 with 5}, {6 with 9}, {14 with 17},and {15 with 18}. All the complementary pairs are used for this square.The complementary pair (16,66) is thrown out. The picture below (Figure A) 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 "Multi-Cross-Over" points are shown as little red stars and is summarized as:

    Picture of a wheel
  8. Figure A
  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
31 45 50
29 44 52
27 43 54
25 42 56
41
26 40 57
28 39 55
30 38 53
32 37 51
A3
31 45 50
29 44 52
27 43 54
25 42 56
6970 71 72 41 10 111213
26 40 57
28 39 55
30 38 53
32 37 51
A4
31 45 50
29 44 52
2735 43 46 54
21 25 42 56 61
6970 71 72 41 10 111213
5826 40 57 24
28 47 39 36 55
30 38 53
32 37 51
A5
31 45 50
29 337 44 74 48 52
202735 43 46 54 62
1921 25 42 56 61 63
6970 71 72 41 10 111213
605826 40 57 2422
5928 47 39 36 55 23
30 49 75 38 834 53
32 37 51
A6
31 0 24 45 7779 81 50
6 29 33 7 44 74 48 52 76
1420 2735 43 46 54 62 68
1519 21 25 42 56 61 6367
6970 71 72 41 10 111213
646058 26 40 57 24 2218
6559 28 47 39 36 55 2317
7330 49 75 38 834 53 9
32 82 8078 37 53 1 51
A6 Border
31 0 24 45 7779 81 50
6 29 337 44 74 48 52 76
1420 2735 43 46 54 62 68
1519 21 25 42 56 61 6367
6970 71 72 41 10 111213
6460 5826 40 57 242218
6559 28 47 39 36 55 2317
7330 49 75 38 834 53 9
32 82 8078 37 5 3 1 51
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
 
82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60
 
23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
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
31 0 24 45 7779 81 50
6 29 33 7 44 74 48 52 76
1420 2735 43 46 54 62 68
1519 21 25 42 56 61 6367
6970 71 72 41 10 111213
646058 26 40 57 24 2218
6559 28 47 39 36 55 2317
7330 49 75 38 834 53 9
32 82 8078 37 53 1 51
A7
4 2 031 45 5081 79 77
7 33 29 6 44 7652 4874
3527 2014 43 68 62 54 46
2521 19 15 42 67 63 6156
7271 70 69 41 13 121110
265860 64 40 18 22 2457
4728 59 65 39 17 23 5536
7549 30 73 38 953 34 8
78 80 8232 37 51 13 5
A8
2521 19 15 42 67 63 6156
3527 2014 43 68 62 54 46
7 33 29 6 44 7652 4874
4 2 031 45 5081 79 77
7271 70 69 41 13 121110
78 80 8232 37 51 13 5
7549 30 73 38 953 34 8
4728 59 65 39 17 23 5536
265860 64 40 18 22 2457

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 D7 of a 9x9 Magic Square Wheel Spoke Shift method. To go to the next 9x9 Part D8.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com