NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part B6

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 "non-Cross-Over" method shown below. For a square with n = 9, there are 23 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 {19,21,23,25} and {32,30,28,26} 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. Table Fe tells us that for n = 9 there are 24 sets that can generate a "non-Cross-Over or terminus" of evenly spaced numbers. The spoke cells are chosen from a group of 23 pairs of evenly spaced numbers. In this exercise we pick the 19th pair (19 → 21 → 23 → 25) and (32 → 30 → 28 → 26) where the non-crossover point between (25,26) is the non-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, 21, 23 and 25 are added, in that order, down to the right and 32, 30, 28 and 26 are added, 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 (9 is paired with 6) 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 {9,6} are 4 units away.
  5. Fill in the next internal square 7x7 by pairing {8 with 5}, {7 with 4}, {34 with 33}, and {16 with 15}.
  6. Fill in the external square 9x9 by pairing {17 with 14}, {24 with 22}, {31 with 27}, {2 with 1}, {3 with -1},and {18 with 20}. All the complementary pairs are used for this square.The complementary pair (29,3) 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 "non-Cross-Over" is shown as a little double arrow 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 50
21 44 52
23 43 54
25 42 56
41
26 40 57
28 39 59
30 38 61
32 37 63
A3
19 45 50
21 44 52
23 43 54
25 42 56
6970 71 72 41 10 111213
26 40 57
28 39 59
30 38 61
32 37 63
A4
19 45 50
21 44 52
239 43 76 54
36 25 42 56 46
6970 71 72 41 10 111213
4726 40 57 35
28 73 39 6 59
30 38 61
32 37 63
A5
19 45 50
21 87 44 78 77 52
34239 43 76 54 48
1636 25 42 56 46 66
6970 71 72 41 10 111213
674726 40 57 3515
4928 73 39 6 59 33
30 74 75 38 45 61
32 37 63
A6
19 17 2431 45 5560 68 50
2 21 8 7 44 78 77 52 80
334 239 43 76 54 48 79
1816 36 25 42 56 46 6664
6970 71 72 41 10 111213
626747 26 40 57 35 1520
8349 28 73 39 6 59 33-1
8130 74 75 38 45 61 1
32 65 5851 37 2722 14 63
A6 Border
19 17 2431 45 5560 68 50
2 21 87 44 78 77 52 80
334 239 43 76 54 48 79
1816 36 25 42 56 46 6664
6970 71 72 41 10 111213
6267 4726 40 57 351520
8349 28 73 39 6 59 33-1
8130 74 75 38 45 61 1
32 65 5851 37 27 22 14 73
-1 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
 
83 ... 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64
 
19 20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
63 62 61 60 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 17 2431 45 5560 68 50
2 21 8 7 44 78 77 52 80
334 239 43 76 54 48 79
1816 36 25 42 56 46 6664
6970 71 72 41 10 111213
626747 26 40 57 35 1520
8349 28 73 39 6 59 33-1
8130 74 75 38 45 61 1
32 65 5851 37 2722 14 63
A7
31 24 1719 45 5068 60 55
7 8 21 2 44 8052 7778
923 343 43 79 48 54 76
2536 16 18 42 64 66 4656
7271 70 69 41 13 121110
264767 62 40 20 15 3557
7328 49 83 39 -1 33 596
7574 30 81 38 161 5 4
51 58 6532 37 63 1422 27
A8
2536 16 18 42 64 66 4656
923 343 43 79 48 54 76
7 8 21 2 44 8052 7778
31 24 1719 45 5068 60 55
7271 70 69 41 13 121110
51 58 6532 37 63 1422 27
7574 30 81 38 161 5 4
7328 49 83 39 -1 33 596
264767 62 40 20 15 3557

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