NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Picture of a wheel

How to Spoke Shift 9x9 Magic Squares-(Part A5)

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 {24,32,30,28} and {23,25,27,29} 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 23rd pair (34 → 32 → 30 → 28) and (23 → 25 → 27 → 29) 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 34, 32, 30 and 28 are added, in that order, down to the right and 23, 25, 27 and 29 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {16,17,18,19} 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 {63,64,65,66} 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 (35 with 36) 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 {15,20} are 5 units away.
  5. Fill in the next internal square 7x7 by pairing {1 with 6}, {7 with 12}, {2 with 3}, and {4 with 5}.
  6. Fill in the external square 9x9 by pairing {24 with 31}, {26 with 33}, {8 with 9}, {10 with 11}, {13 with 14},and {21 with 22}. All the complementary pairs are used for this square. 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. 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.
  9. 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
19 45 80
16 44 77
13 43 74
10 42 71
41
11 40 72
8 39 69
5 38 66
2 37 63
A3
34 45 59
32 44 57
30 43 55
28 42 53
6364 65 66 41 16 1718 19
29 40 54
27 39 52
25 38 50
23 37 48
A4
34 45 59
32 44 57
3015 43 62 55
36 28 42 53 46
6364 65 66 41 16 1718 19
4729 40 54 35
27 67 39 20 52
25 38 50
23 37 48
A5
34 45 59
32 17 44 70 76 57
33015 43 62 55 79
536 28 42 53 46 77
6364 65 66 41 16 1718 19
784729 40 54 354
8027 67 39 20 52 2
25 81 75 38 126 50
23 37 48
A6
34 24 268 45 7349 51 59
11 32 1 7 44 70 76 57 71
143 3015 43 62 55 79 68
225 36 28 42 53 46 7760
6364 65 66 41 16 1718 19
617847 29 40 54 35 421
6980 27 67 39 20 52 213
7225 81 75 38 126 50 10
23 58 5674 37 933 31 48
A6 Border
34 24 268 45 7349 51 59
11 32 17 44 70 76 57 71
143 3015 43 62 55 79 68
225 36 28 42 53 46 7760
6364 65 66 41 16 171819
6178 4729 40 54 35421
6980 27 67 39 20 52 213
7225 81 75 38 126 50 10
23 58 5674 37 9 33 31 48
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
34 24 268 45 7349 51 59
11 32 1 7 44 70 76 57 71
143 3015 43 62 55 79 68
225 36 28 42 53 46 7760
6364 65 66 41 16 1718 19
617847 29 40 54 35 421
6980 27 67 39 20 52 213
7225 81 75 38 126 50 10
23 58 5674 37 933 31 48
A7
8 16 2434 45 5951 49 73
7 1 32 11 44 71 57 76 70
1530 314 43 68 79 55 62
2836 5 22 42 60 77 4653
6665 64 63 41 19 181716
294778 61 40 21 4 3554
6727 80 69 39 13 2 5220
7581 25 72 38 1050 6 12
74 56 5823 37 48 3133 9
A8
2836 5 22 42 60 77 4653
1530 314 43 68 79 55 62
7 1 32 11 44 71 57 76 70
8 16 2434 45 5951 49 73
6665 64 63 41 19 181716
74 56 5823 37 48 3133 9
7581 25 72 38 1050 6 12
6727 80 69 39 13 2 5220
294778 61 40 21 4 3554

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 D.
To go back to 9x9 Part B.
Go back to homepage.


Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com