NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part D2
How to Spoke Shift 7x7 Magic Squares
A magic square is an arrangement of numbers 1,2,3,... n^{2} 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
n^{2} + 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(n^{2} + 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 "MultiCrossOver" method shown
below. For a square with n = 7, there are 16 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 {1,3,5} and {2,4,6} pairs from the 7x7 Part D1 except for the reversal of some of the diagonal numbers.
(Total reversal is not possible since it changes the order or structure of the middle column).
The connectivity of all the numbers used in the square is also shown in 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 
A 7x7 Magic Square Using the Pairs {1,3,5} and {2,4,6} in Reversed Fashion
 The center column is filled with the group of numbers ½
(n^{2}n+2) to ½(n^{2}+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).
 12 pairs are left with which to construct the spoke and fill in the nonspoke cells. Table F_{g}
tells us that for n = 7 there are 16 sets that can generate a
"multiCrossOver or terminus" of evenly spaced numbers.
The spoke cells are then chosen from a group of 16 pairs of evenly spaced numbers. For this exercise we pick the
the 1^{st} "multiCrossOver pairs" (5 → 3 → 1) and
(6 → 4 → 2) where 3 crossover points are present at (2,3), (3,4) and (4,5).
The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A2. The numbers 1, 3 and 5 are added,
in that order, down to the right and 2, 4 and 6 are added, in that order, up right as shown.
 This is followed by adding the pairs {22,21,20} to the center row with 22 to the right of 25, adding the next numbers consecutively to the right hand side of
the square and finishing of with their complements {70,71,72} to the left of 25 (Square A3).
 To fill up the rest of the square work with the internal square first, i.e., 5x5 where (20 is paired with 21) and (12 with 15) 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 (12,15) are 4 units away.
 Fill in the external square 7x7 by pairing (18 with 19), (16 with 17), (7 with 10), and (8 with 11). The complementary pair (9,41),(13,37) and (14,36)
are thrown out. The picture below shows the physical connectivity.
 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 nonspoke
numbers and the "multiCrossOver or terminus" is shown as little stars, is summarized as:
Figure A
 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.
 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.

⇒ 
A2
5   
28   
44 
 3  
27   46 

  1  26 
49   
  
25  
 
  2 
24  49 
 
 4  
23  
47  
6   
22   
45 

⇒ 
A3
5   
28   
44 
 3  
27   46 

  1  26 
49   
70  71  72 
25  22 
21  20 
  2 
24  49 
 
 4  
23  
47  
6   
22   
45 

 ⇒ 
A4
5   
28   
44 
 3  20 
27  29  46 

 12  1  26 
48  38  
70  71  72 
25  22 
21  20 
 35  2 
24  49 
15  
 4  30 
23  21 
47  
6   
22   
45 

 ⇒ 
A5
5  18  16 
28  33  31 
44 
7  3  20 
27  29  46 
43 
8  12  1 
26  48 
38  42 
70  71  72 
25  22 
21  20 
39  35  2 
24  49 
15  11 
40  4  30 
23  21 
47  10 
6  32  34 
22  17  19 
45 

⇒ 
A5 Border
5  18  16 
28  33  31 
44 
7  3  20 
27  29  46 
43 
8  12  1 
26 
48  38  42 
70  71  72 
25  22 
21  20 
39  35  2 
24  49 
15  11 
40  4  30 
23  21 
47  10 
6  32  34 
22  17  19 
45 

22  21  20 
...  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 
72  71  70 
...  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.
 Take square A5 and transpose (column 1 with column 3) and (column 5 with column 7) to get Square A6.
 Take square A6 and transpose (row 1 and row 3) and (row 5 with row 7) to get Square A7.
 In a sense A5 has been imploded or everted into A7, i.e., A5 and A7 below are opposites.
A5
5  18  16 
28  33  31 
44 
7  3  20 
27  29  46 
43 
8  12  1 
26  48 
38  42 
70  71  72 
25  22 
21  20 
39  35  2 
24  49 
15  11 
40  4  30 
23  21 
47  10 
6  32  34 
22  17  19 
45 

⇒ 
A6
16  18  5 
28  44  31 
33 
20  3  7 
27  43  46 
29 
1  12  8 
26  42 
38  48 
72  71  70 
25  20 
21  22 
2  35  39 
24  11 
15  49 
30  4  40 
23  10 
47  21 
34  32  6 
22  45  19 
17 

⇒ 
A7
1  12  8 
26  42 
38  48 
20  3  7 
27  43  46 
29 
16  18  5 
28  44  31 
33 
72  71  70 
25  20 
21  22 
34  32  6 
22  45  19 
17 
30  4  40 
23  10 
47  21 
2  35  39 
24  11 
15  49 

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
nonspoke cells which appears to be more difficult to do.
This completes part D2 of a 7x7 Magic Square Wheel Spoke Shift method. To see the next 7x7 Part D3.
Go back to homepage.
Copyright © 2014 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com