NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part C4

Picture of a wheel

How to Spoke Shift 7x7 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.

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 = 7, there are 10 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} and {9,6,3} pairs along with the connectivity of all the numbers used for the squares (Figure A).

The new magic squares with n = 7 are constructed as follows using a complimentary table as a guide.


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
50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

The squares in this section are produced by reversing the diagonal numbers from the previous section of 7x7 squares . Originally the diagonal pairs are listed as {2,5,8} and {9,6,3} and reversal produces {8,5,2} and {3,6,9} which are still capable of forming magic squares without producing duplicates. In another case it will be shown that this reversal can not be used and an alternative method is required.

A 7x7 Magic Square Using the Pairs {2,5,8} and {9,6,3}

  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 = 7. For a 7x7 square the numbers in the center column correspond to 22 → 23 → 24 → 25 → 26 → 27 → 28 starting from the bottom (Square A1).
  2. 12 pairs are left with which to construct the spoke and fill in the non-spoke cells. Table Ff tells us that for n = 7 there are 12 sets that can generate a "multi-Cross-Over or terminus" of evenly spaced numbers. The spoke cells are then chosen from a group of 12 pairs of evenly spaced numbers. For this exercise we pick the the 2nd "multi-Cross-Over pairs" (2 → 5 → 8) and (9 → 6 → 3) where three crossover points are present at (3,5), (5,6) and (6,8). The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A2. The numbers 2, 5 and 8 are added, in that order, down to the right and 3, 6 and 9 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {-8,-7,-6} to the center row with -8 to the right of 25, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {56,57,58} to the left of 25 (Square A3).
  4. To fill up the rest of the square work with the internal square first, i.e., 5x5 where (20 is paired with 21) and (4 with 11) along with their complements in the same row or column to form Square A4. Note that (20,21) are adjacent on the complementary table while (4,11) are 9 units away.
  5. Fill in the external square 7x7 by pairing (13 with 14), (15 with 16), (17 with 10), and (19 with 12). The complementary pair (1,49),(7,43) and (18,32) are thrown out. The picture below shows the physical connectivity.
  6. The portion of the complementary table (just the top set of numbers since the same applies to the bottom set) showing all the connectivities of the non-spoke numbers and the "multi-Cross-Over or terminus" is shown as little stars, is summarized as:

    Picture of a wheel
    Figure A
  7. The result of these operations is a wheel with a shifted spoke where the numbers in the diagonal of the regular wheel 22 → 23 → 24 → 25 → 26 → 27 → 28 have been transposed or shifted to a column.
  8. The square that is produced via this method is a border square, since the 3x3 square has an S = 75, the 5x5 has an S = 125 while the 7x7 has an S = 175. These border squares are shown in Square A5.
A1
28
27
26
25
24
23
22
A2
2 28 47
5 27 44
8 26 41
25
9 24 42
6 23 45
3 22 48
A3
2 28 47
5 27 44
8 26 41
56 5758 25 -8 -7-6
9 24 42
6 23 45
3 22 48
A4
2 28 47
5 20 2729 44
11 8 26 41 39
56 5758 25 -8 -7-6
46 9 24 42 4
6 30 23 21 45
3 22 48
A5
2 13 15 28 3436 47
17 5 20 27 29 44 33
19 11 8 26 41 39 31
56 5758 25 -8 -7-6
3846 9 24 42 4 12
406 30 23 21 45 10
3 37 35 22 1614 48
A5 Border
2 13 15 28 3436 47
17 5 20 27 29 44 33
19 11 8 26 41 39 31
56 5758 25 -8 -7-6
3846 9 24 42 4 12
406 30 23 21 45 10
3 37 35 22 1614 48
-8 -7 -6 ... 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
585756 ... 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

Conversion of the 7x7 into its transposed opposite

Generation of a 7x7 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 A5 and transpose (column 1 with column 3) and (column 5 with column 7) to get Square A6.
  2. Take square A6 and transpose (row 1 and row 3) and (row 5 with row 7) to get Square A7.
  3. In a sense A5 has been imploded or everted into A7, i.e., A5 and A7 below are opposites.
A5
2 13 15 28 3436 47
17 5 20 27 29 44 33
19 11 8 26 41 39 31
56 5758 25 -8 -7-6
3846 9 24 42 4 12
406 30 23 21 45 10
3 37 35 22 1614 48
A6
15 13 2 28 4736 34
20 5 17 27 33 44 29
8 11 19 26 31 39 41
58 5756 25 -6 -7-8
946 38 24 12 4 42
306 40 23 10 45 21
35 37 3 22 4814 16
A7
8 11 19 26 31 39 41
20 5 17 27 33 44 29
15 13 2 28 4736 34
58 5756 25 -6 -7-8
35 37 3 22 4814 16
306 40 23 10 45 21
946 38 24 12 4 42

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 first variant of a 7x7 Magic Square Wheel Spoke Shift method. To see the next 7x7 Part C5.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com