NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part H3

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 {2,3,4,6} and {5,7,8,9} 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 {2,3,4,6} and {5,7,8,9}

  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 2nd pair (2 → 3 → 4 → 6) and (5 → 7 → 8 → 9) where the crossover point (5,6) 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 9, 8, 7 and 5 are added, in that order, i.e., in reverse, down to the right and 2, 3, 4 and 6 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {-30,-29,-28,-27} to the center row with -30 to the right of 41, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {109,110,111,112} 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 (31 is paired with 36) and (29 with 28) along with their complements in the same row or column to form Square A4. Note that {29,28} are adjacent on the complementary table while {31,36} are 6 units away.
  5. Fill in the next internal square 7x7 by pairing {10 with 14}, {11 with 15}, {35 with 34}, and {33 with 32}.
  6. Fill in the external square 9x9 by pairing {12 with 16}, {13 with 17}, {27 with 30}, {22 with 21}, {24 with 23},and {26 with 25}. The complementary pair (1,81), (18,64), (19,63) and (20,62) are 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
9 45 80
8 44 79
7 43 78
5 42 76
41
6 40 77
4 39 75
3 38 74
2 37 73
A3
9 45 80
8 44 79
7 43 78
5 42 76
109110 111 112 41 -30 -29-28-27
6 40 77
4 39 75
3 38 74
2 37 73
A4
9 45 80
8 44 79
731 43 46 78
29 5 42 76 53
109110 111 112 41 -30 -29-28-27
546 40 77 28
4 51 39 36 75
3 38 74
2 37 73
A5
9 45 80
8 1011 44 67 68 79
35731 43 46 78 47
3329 5 42 76 53 49
109110 111 112 41 -30 -29-28-27
50546 40 77 2832
484 51 39 36 75 34
3 72 71 38 1514 74
12 37 73
A6
9 12 1327 45 5265 66 80
22 8 10 11 44 67 68 79 60
2435 731 43 46 78 47 58
2633 29 5 42 76 53 4956
109110 111 112 41 -30 -29-28-27
575054 6 40 77 28 3225
5948 4 51 39 36 75 3423
613 72 71 38 1514 74 21
2 70 6955 37 3017 16 73
A6 Border
9 12 1327 45 5265 66 80
22 8 1011 44 67 68 79 60
2435 731 43 46 78 47 58
263329 5 42 76 53 4956
109110 111 112 41 -30 -29-28-27
575054 6 40 77 283225
59484 51 39 36 75 3423
613 72 71 38 1514 74 21
2 70 6955 37 30 17 16 73
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
 
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
9 12 1327 45 5265 66 80
22 8 10 11 44 67 68 79 60
2435 731 43 46 78 47 58
2633 29 5 42 76 53 4956
109110 111 112 41 -30 -29-28-27
575054 6 40 77 28 3225
5948 4 51 39 36 75 3423
613 72 71 38 1514 74 21
2 70 6955 37 3017 16 73
A7
27 13 129 45 8066 65 52
11 10 8 22 44 60 79 68 67
317 3524 43 58 47 78 46
529 33 26 42 56 49 5376
112111 110 109 41 -27 -28-29-30
65450 57 40 25 32 2877
514 48 59 39 23 34 7536
7172 3 61 38 2174 14 15
55 69 702 37 73 1617 30
A8
529 33 26 42 56 49 5376
317 3524 43 58 47 78 46
11 10 8 22 44 60 79 68 67
27 13 129 45 8066 65 52
112111 110 109 41 -27 -28-29-30
55 69 702 37 73 1617 30
7172 3 61 38 2174 14 15
514 48 59 39 23 34 7536
65450 57 40 25 32 2877

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. To go forward to 9x9 Part H4.
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Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com