NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Picture of a wheel

How to Spoke Shift 7x7 Magic Squares-(Part G)

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 new method 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.

This method consists of transposing rows and columns around to generate a magic square where the spoke numbers have been inverted. The method generates non-border squares where only the external square is magic.It differs from the previous method in the addition of the central colum is done in a reverse fashion. Again the internal squares are 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 = 7, there are 14 sets of pairs. These pairs and their complements make up entries to the diagonal cells. A diagram of the {12,14,16} and {9,11,13} connectivity is shown below in Figure A.

The new magic squares with n = 7 are constructed as follows using a complimentary table as a guide. In this method the center column is now added in the reverse manner than in the first method discussed for these 7x7 squares.


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
  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 top 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 top (Square A1).
  2. 12 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 14 pairs of evenly spaced numbers. In this exercise we pick the 9th pair (12 → 14 → 16) and (13 → 11 → 9) where the crossover point (12,13) 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 12, 14 and 16 are added, in that order, down to the right and 13, 11 and 9 are added, in that order, up right as shown.
  3. This is followed by adding the pairs {2,1,0} to the center row with 0 to the right of 25, adding the next numbers consecutively to the right hand side of the square and finishing of with their complements {50,49,48} 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 (3 is paired with 8) and (4 with 5) along with their complements in the same row or column to form Square A4.
  5. Fill in the external square 7x7 by pairing {18 with 19}, {20 with 21}, {15 with 17}, and {8 with 10}. The complementary pair {7,43} 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 the connectivity of the non-spoke numbers and the "Cross-Over" is shown as a little red cross, is summarized as:

    Picture of numbers
  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 not a border square, since the internal 3x3 and 5x5 squares are not magic as is the 7x7 which has an S = 175.
A1
22
23
24
25
26
27
28
A2
12 22 41
14 23 39
16 24 37
25
9 26 38
11 27 36
13 28 34
A3
12 22 41
14 23 39
16 26 37
48 49 50 25 0 1 2
9 26 38
11 27 36
13 28 34
A4
12 22 41
14 3 23 46 39
45 16 24 37 5
48 49 50 25 0 1 2
6 9 26 38 44
11 47 27 4 36
13 28 34
A5
12 18 21 22 3031 41
17 14 3 23 46 39 33
40 45 16 24 37 5 8
48 49 50 25 0 1 2
106 9 26 38 44 42
35 11 47 27 4 36 15
13 32 29 28 2019 34
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

Conversion of the 7x7 variant 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
12 18 21 22 3031 41
17 14 3 23 46 39 33
40 45 16 24 37 5 8
48 49 50 25 0 1 2
106 9 26 38 44 42
35 11 47 27 4 36 15
13 32 29 28 2019 34
A6
21 18 12 22 4131 30
3 14 17 23 33 39 46
16 45 40 24 8 5 37
50 49 48 25 2 1 0
9 6 10 26 42 44 38
47 11 35 27 15 36 4
29 32 13 28 3419 20
A7
16 45 40 24 8 5 37
3 14 17 23 33 39 46
21 18 12 22 4131 30
50 49 48 25 2 1 0
29 32 13 28 3419 20
47 11 35 27 15 36 4
9 6 10 26 42 44 38

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 7x7 Magic Square Wheel Spoke Shift method using a variant reversion of the central column numbers.

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Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com