NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part D8

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 {31,29,27,25} and {32,30,28,26} pairs along with the connectivity of all the numbers used for the squares (Figure A) but in a partially reversed fashion, i.e., {31,27,29,25} and {32,28,30,26} since a totally reversed fashion produces duplicates.

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 {31,29,25,25} and {32,30,28,26} in Partially Reversed Fashion

  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 (31 → 29 → 27 → 25) and (32 →30 → 28 → 26) 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 are added in partially reversed fashion, i.e., 31, 27, 29 and 25 in that order, down to the right and 32, 28, 30 and 26 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 (15 with 22) 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 {15,22} are 7 units away.
  5. Fill in the next internal square 7x7 by pairing {23 with 24}, {33 with 34}, {2 with 3}, and {0 with 1}.
  6. Fill in the external square 9x9 by pairing {4 with 5}, {6 with 7}, {8 with 9}, {14 with 17}, {16 with 19},and {18 with 21}. All the complementary pairs are used for this square.The complementary pair (20,62) 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
27 44 54
29 43 52
25 56 72
41
26 40 57
30 39 53
28 38 55
32 37 51
A3
31 45 50
27 44 54
29 43 52
25 56 72
6970 71 72 41 10 111213
26 40 57
30 39 53
28 38 55
32 37 51
A4
31 45 50
27 44 54
2935 43 46 52
15 25 42 56 67
6970 71 72 41 10 111213
6026 40 57 22
30 47 39 36 53
28 38 55
32 37 51
A5
31 45 50
27 2333 44 48 58 54
22935 43 46 52 80
015 25 42 56 67 82
6970 71 72 41 10 111213
816026 40 57 221
7930 47 39 36 53 3
28 59 49 38 3424 55
32 37 51
A6
31 4 68 45 7375 77 50
14 27 23 33 44 48 5854 68
162 2935 43 46 52 80 66
180 15 25 42 56 67 8264
6970 71 72 41 10 111213
618160 26 40 57 22 121
6379 30 47 39 36 53 319
6528 59 49 38 3424 55 17
32 78 7674 37 97 5 51
A6 Border
31 4 68 45 7375 77 50
14 27 2333 44 48 58 54 68
162 2935 43 46 42 80 66
180 15 25 42 56 67 8264
6970 71 72 41 10 111213
6181 6026 40 57 22121
6379 30 47 39 36 53 319
6528 59 49 38 3424 55 17
32 78 7674 37 9 7 5 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 4 68 45 7375 77 50
14 27 23 33 44 48 5854 68
162 2935 43 46 52 80 66
180 15 25 42 56 67 8264
6970 71 72 41 10 111213
618160 26 40 57 22 121
6379 30 47 39 36 53 319
6528 59 49 38 3424 55 17
32 78 7674 37 97 5 51
A7
8 6 431 45 5077 75 73
33 23 27 14 44 6854 5848
3529 216 43 66 80 52 46
2515 0 18 42 64 82 6756
7271 70 69 41 13 121110
266081 61 40 21 1 2257
4730 79 63 39 19 3 5336
4959 28 65 38 1755 24 34
74 76 7832 37 51 57 9
A8
2515 0 18 42 64 82 6756
3529 216 43 66 80 52 46
33 23 27 14 44 6854 5848
8 6 431 45 5077 75 73
7271 70 69 41 13 121110
74 76 7832 37 51 57 9
4959 28 65 38 1755 24 34
4730 79 63 39 19 3 5336
266081 61 40 21 1 2257

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 D8 of a 9x9 Magic Square Wheel Spoke Shift method. Go back to homepage.


Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com