NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part A5

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 the number 0 and its complement are now part of the square. Since the number of cells in an nxn magic squares is n then a complementary pair (not containing 0) is not used in generating the square. The use of 0 is a requirement because the use of numbers from 1 to n cannot generate this type of magic 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 "Cross-Over" method shown below. For a square with n = 9, there are 19 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 {2,5,8,11} and {19,16,13,10} 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
  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. The spoke cells are chosen from a group of 19 pairs of evenly spaced numbers. In this exercise we pick the 2nd pair (2 → 5 → 8 → 11) and (19 → 16 → 13 → 10) where the crossover point between (10,11) 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 19, 16, 13 and 10 are added, in that order, down to the right and 2, 5, 8 and 11 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {-20,-19,-18,-17} 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 {99,100,101,102} 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 (18 is paired with 25) 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 {18,25} are 8 units away.
  5. Fill in the next internal square 7x7 by pairing {17 with 24}, {15 with 22}, {27 with 28}, and {29 with 30}.
  6. Fill in the external square 9x9 by pairing {14 with 21}, {3 with 9}, {12 with 20}, {33 with 34}, {31 with 32},and {6 with 7}. All the complementary pairs are used for this square.The complementary pairs (1,81),(4,78), (23,59) and (26,56) 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 "Cross-Over" is shown as a little red cross 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
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
19 45 80
16 44 77
13 43 74
10 42 71
99100 101 102 41 -20 -19-18-17
11 40 72
8 39 69
5 38 66
2 37 63
A4
19 45 80
16 44 77
1318 43 57 74
36 10 42 71 46
99100 101 102 41 -20 -19-18-17
4711 40 72 35
8 64 39 25 69
5 38 66
2 37 63
A5
19 45 80
16 1715 44 60 58 77
281318 43 57 74 54
3036 10 42 71 46 52
99100 101 102 41 -20 -19-18-17
534711 40 72 3529
558 64 39 25 69 27
5 65 67 38 2224 66
2 37 63
A6
19 14 312 45 6273 61 80
34 16 17 15 44 60 58 77 48
3228 1318 43 57 74 54 50
730 36 10 42 71 46 5275
99100 101 102 41 -20 -19-18-17
765347 11 40 72 35 296
5155 8 64 39 25 69 2731
495 65 67 38 2224 66 33
2 68 7970 37 209 21 63
A6 Border
19 14 312 45 6273 61 80
34 16 1715 44 60 58 77 48
3228 1318 43 57 74 54 50
730 36 10 42 71 46 5275
99100 101 102 41 -20 -19-18-17
7653 4711 40 72 35296
5155 8 64 39 25 69 2731
495 65 67 38 2224 66 33
2 68 7970 37 20 9 21 63
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
19 14 312 45 6273 61 80
34 16 17 15 44 60 58 77 48
3228 1318 43 57 74 54 50
730 36 10 42 71 46 5275
99100 101 102 41 -20 -19-18-17
765347 11 40 72 35 296
5155 8 64 39 25 69 2731
495 65 67 38 2224 66 33
2 68 7970 37 209 21 63
A7
12 3 1419 45 8061 73 62
15 17 16 34 44 48 77 58 60
1813 2832 43 50 54 74 57
1036 30 7 42 75 52 4671
102101 100 99 41 -17 -18-19-20
114753 76 40 6 29 3572
648 55 51 39 31 27 6925
6765 5 49 38 3366 24 22
70 79 682 37 63 219 20
A8
1036 30 7 42 75 52 4671
1813 2832 43 50 54 74 57
15 17 16 34 44 48 77 58 60
12 3 1419 45 8061 73 62
102101 100 99 41 -17 -18-19-20
70 79 682 37 63 219 20
6765 5 49 38 3366 24 22
648 55 51 39 31 27 6925
114753 76 40 6 29 3572

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