NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part C8

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 26 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 {3,6,9,12} and {4,7,10,13} 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

The squares in this section are produced by reversing the diagonal numbers from the previous section of 9x9 squares . Originally the diagonal pairs were listed as {12,9,6,3} and {13,10,7,4} and reversal produces {3,6,9,12} and {4,7,10,13}.

  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 Ff tells us that for n = 9 there are 26 sets that can generate a "Multi-Cross-Over or terminus" of evenly spaced numbers. The spoke cells are chosen from a group of 26 pairs of evenly spaced numbers. In this exercise we pick the 3rd pair (3 → 6 → 9 → 12) and (13 → 10 → 7 → 4) where 5 Multi-Crossover points are present at (4,6), (6,7), (7,9), (9,10) and (10,12). The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A2. The numbers 3, 6, 9 and 12 are added, in that order, down to the right and 4, 7, 10 and 13 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {-16,-15,-14,-13} to the center row with -16 to the right of 41, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {95,96,97,98} 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 (33 is paired with 34) and (14 with 19) 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 {24,17} are 8 units away.
  5. Fill in the next internal square 7x7 by pairing {31 with 32}, {29 with 30}, {23 with 16}, and {22 with 15}.
  6. Fill in the external square 9x9 by pairing {27 with 28}, {25 with 26}, {-1 with 0}, {8 with 1}, {14 with 2},and {20 with 15}. All the complementary pairs are used for this square.The complementary pairs (5,77),(11,71), (19,63) and (21,61) are 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
3 45 78
6 44 75
9 43 72
12 42 69
41
13 40 70
10 39 73
7 38 76
4 37 79
A3
3 45 78
6 44 75
9 43 72
12 42 69
9596 97 98 41 -16 -15-14-13
13 40 70
10 39 73
7 38 76
4 37 79
A4
3 45 78
6 44 75
933 43 48 72
24 12 42 69 58
9596 97 98 41 -16 -15-14-13
6513 40 70 17
10 49 39 34 73
7 38 76
4 37 79
A5
3 45 78
6 3129 44 52 50 75
23933 43 48 72 59
2224 12 42 69 58 60
9596 97 98 41 -16 -15-14-13
676513 40 70 1715
661049 39 34 73 16
7 51 53 38 3032 76
4 37 79
A6
3 27 2535 45 3656 54 78
8 6 31 29 44 52 50 75 74
1423 933 43 48 72 59 68
2022 24 12 42 69 58 6062
9596 97 98 41 -16 -15-14-13
646765 13 40 70 17 1518
8066 10 49 39 34 73 162
817 51 53 38 3032 76 1
4 55 5747 37 4626 28 79
A6 Border
3 27 2535 45 3656 54 78
8 6 3129 44 52 50 75 74
1423 933 43 48 72 59 68
2022 24 12 42 69 58 6062
9596 97 98 41 -16 -15-14-13
6467 6513 40 70 171518
8066 10 49 39 34 73 162
817 51 53 38 3032 76 1
4 55 5747 37 46 26 28 79
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
 
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
3 27 2535 45 3656 54 78
8 6 31 29 44 52 50 75 74
1423 933 43 48 72 59 68
2022 24 12 42 69 58 6062
9596 97 98 41 -16 -15-14-13
646765 13 40 70 17 1518
8066 10 49 39 34 73 162
817 51 53 38 3032 76 1
4 55 5747 37 4626 28 79
A7
35 25 273 45 7854 56 36
29 31 6 8 44 7475 5052
339 2314 43 68 59 72 48
1224 22 20 42 62 60 5869
9897 96 95 41 -13 -14-15-16
136567 64 40 18 15 1770
4910 66 80 39 2 16 7334
5351 7 81 38 176 32 30
47 57 554 37 79 2826 46
A8
1224 22 20 42 62 60 5869
339 2314 43 68 59 72 48
29 31 6 8 44 7475 5052
35 25 273 45 7854 56 36
9897 96 95 41 -13 -14-15-16
47 57 554 37 79 2826 46
5351 7 81 38 176 32 30
4910 66 80 39 2 16 7334
136567 64 40 18 15 1770

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 C of a 9x9 Magic Square Wheel Spoke Shift method. To go to the next 9x9 Part C9.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com