NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part H4

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 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 "Single-Cross-Over" method shown below. For a square with n = 9, there are 29 sets of pairs. These pairs and their complements make up entries to the diagonal cells. A diagram of the {27,28,29,31} and {30,32,33,34} 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

A 9x9 Magic Square Using the Pairs {27,28,29,31} and {30,32,33,34}

  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 Fh tells us that for n = 9 there are 29 sets that can generate a "Single-Cross-Over" 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 27th pair (27 → 28 → 29 → 31) and (30 → 32 → 33 → 34) where the crossover point (30,31) is the "Single-Cross-Over". Note that one set of numbers goes to the right the other to the left. 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, 33, 32 and 30 are added, in that order, i.e., in reverse, down to the right and 27, 28, 29 and 31 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {20,21,22,23} to the center row with 20 to the right of 41, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {59,60,61,62} 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 (6 is paired with 11) and (36 with 35) along with their complements in the same row or column to form Square A4. Note that {36,35} are adjacent on the complementary table while {6,11} are 6 units away.
  5. Fill in the next internal square 7x7 by pairing {15 with 19}, {14 with 18}, {25 with 24}, and {9 with 8}.
  6. Fill in the external square 9x9 by pairing {13 with 17}, {12 with 16}, {7 with 10}, {1 with 0}, {3 with 2},and {5 with 4}. The complementary pair (26,56) is thrown out. 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 "Single-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
34 45 55
33 44 54
32 43 53
30 42 51
41
31 40 52
29 39 50
28 38 49
27 37 48
A3
34 45 55
33 44 54
32 43 53
30 42 51
5960 61 62 41 20 212223
31 40 52
29 39 50
28 38 49
27 37 48
A4
34 45 55
33 44 54
326 43 71 53
36 30 42 51 46
5960 61 62 41 20 212223
4731 40 52 35
29 76 39 11 50
28 38 49
27 37 48
A5
34 45 55
33 1514 44 64 63 54
25326 43 71 53 57
936 30 42 51 46 73
5960 61 62 41 20 212223
744731 40 52 358
5829 76 39 11 50 24
28 67 68 38 1819 49
27 37 48
A6
34 13 127 45 7266 65 55
1 33 15 14 44 64 63 54 81
325 326 43 71 53 57 79
59 36 30 42 51 46 7377
5960 61 62 41 20 212223
787447 31 40 52 35 84
8058 29 76 39 11 50 242
8228 67 68 38 1819 49 0
27 69 7075 37 1016 17 48
A6 Border
34 13 127 45 7266 65 55
1 33 1514 44 64 63 54 81
325 326 43 71 53 57 79
5936 30 42 51 46 7377
5960 61 62 41 20 212223
787447 31 40 52 3584
805829 76 39 11 50 242
8228 67 68 38 1819 49 0
27 69 7075 37 10 16 17 48
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
 
82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56
 
27 28 29 30 3132 33 34 35 36 37 38 39 40
41
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 13 127 45 7266 65 55
1 33 15 14 44 64 63 54 81
325 326 43 71 53 57 79
59 36 30 42 51 46 7377
5960 61 62 41 20 212223
787447 31 40 52 35 84
8058 29 76 39 11 50 242
8228 67 68 38 1819 49 0
27 69 7075 37 1016 17 48
A7
7 12 1334 45 5565 66 72
14 15 33 1 44 81 54 63 64
632 253 43 79 57 53 71
3036 9 5 42 77 73 4651
6261 60 59 41 23 222120
314774 78 40 4 8 3552
7629 58 80 39 2 24 5011
6867 28 82 38 049 19 18
75 70 6927 37 48 1716 10
A8
3036 9 5 42 77 73 4651
632 253 43 79 57 53 71
14 15 33 1 44 81 54 63 64
7 12 1334 45 5565 66 72
6261 60 59 41 23 222120
75 70 6927 37 48 1716 10
6867 28 82 38 049 19 18
7629 58 80 39 2 24 5011
314774 78 40 4 8 3552

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 the Part A of a 9x9 Magic Square Wheel Spoke Shift method.
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Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com