NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part C1
How to Spoke Shift 5x5 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 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 numbers 0, 1,2 etc and their complements may
may now be part of the square. Since the number of cells in an nxn magic squares is n then a complementary pair
(not containing 0 or any minus number) is not used in generating the square. The use of these other numbers is a requirement because the use of numbers from 1
to n may not be enough to 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.
In addition, the diagonal pairs are obtained from the complementary table using what I call a "CrossOver" method shown
below. For a square with n = 5, there are six sets of pairs. These are the (a) {1,4} and {5,2}, (b) {2,5} and {6,3},
(c) {3,6} and {7,4}, (d) {4,7} and {8,5} and (e) {5,8} and {9,6} and (e) {6,9} and {10,7} and are tabulated as such as
the group of 6 pairs. These pairs and their complements make up entries to the diagonal cells.
A diagram of the {2,4} and {5,7} connectivity is shown below in Figure A.
The new magic squares with n = 5 are constructed as follows using a complimentary table as a guide.
1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 13 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
A 5x5 Magic Square Using the Pairs {1,4} and {5,2}
 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 = 5.
For a 5x5 square the numbers in the center column correspond to 11 → 12 → 13 → 14 → 15 starting from the bottom (Square A1).
Other series are possible but for these particular 5x5 squares we will concentrate only on these for now.
 7 pairs are left with which to construct the spoke and fill in the nonspoke cells. Table F_{f}
tells us that for n = 5 there are 6 sets that can generate a
"CrossOver" of evenly spaced numbers.
The spoke cells are then chosen from a group of 6 pairs of evenly spaced numbers. For this exercise we pick the
the 1^{st} "CrossOver pairs" (1 → 4) and
(5 → 2) with one crossover point at adjacent points 2 and 4 at the terminus.
The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A2. The numbers 4 and 1 are added, in that order,
down to the right and 5 and 2 are added, in that order, up right as shown.
 This is followed by adding the pairs {7,6} to the center row with 7 to the right of 13, adding the next numbers consecutively to the right hand side of the square and
finishing of with their complements {35,36} to the left of 13 (Square A3).
 To fill up the rest of the square (6 is paired with 7) and (3 with 8) along with their complements in the same row or column to finish off Square A4.
Note that {6,7} are adjacent on the complementary table while {3,8} are 6 units away. The complementary pairs (9,17) and (10,16) are thrown out.
 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 nonspoke
numbers and the "CrossOver or terminus" is shown as a little red star, 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 11 → 12 → 13 → 14 → 15.
have been transposed or shifted to a column.
 The square that is produced via this method (A4) is a border square, since the internal 3x3 square has a Sum of 39 while the external has a Sum of 65.

⇒ 
A2
4  
15  
21 
 1 
14  24 

 
13  

 2 
12  25 

5  
11  
22 

⇒ 
A3
4  
15  
21 
 1 
14  24 

35  36 
13  7 
6 
 2 
12  25 

5  
11  
22 

 ⇒ 
A4
4  6 
15  19 
21 
3  1 
14  24 
23 
35  36 
13  7 
6 
18  2 
12  25 
8 
5  20 
11  7 
22 

7  6 
...  1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 13 
36  35 
...  25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
Conversion of the 5x5 into its transposed opposite
Generation of a 5x5 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 A4 and transpose (column 1 with column 2) and (column 4 with column 5) to get Square A5.
 Take square A5 and transpose (row 1 and row 2) and (row 4 with row 5) to get Square A6.
 In a sense A4 has been imploded or everted into A6, i.e., A4 and A6 below are opposites.
A4
4  6 
15  19 
21 
3  1 
14  24 
23 
35  36 
13  7 
6 
18  2 
12  25 
8 
5  20 
11  7 
22 

⇒ 
A5
6  4 
15  21 
19 
1  3 
14  23 
24 
36  35 
13  6 
7 
2  18 
12  8 
25 
20  5 
11  22 
7 

⇒ 
A6
1  3 
14  23 
24 
6  4 
15  21 
19 
36  35 
13  6 
7 
20  5 
11  22 
7 
2  18 
12  8 
25 

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.
A Variant of Square A4
 The crossover points for a 5x5 may be inverted on a square to produce an alternative magic square. Inverting the
"CrossOver pairs" {4,1} and {2,5} and going thru the requisite steps produces the square A41.
 In this case the complementary pairs {8,18} and {9,17} are thrown out.
 Again this square is a border square.

⇒ 
A21
1  
15  
24 
 4 
14  21 

 
13  

 5 
12  22 

2  
11  
25 

⇒ 
A31
1  
15  
24 
 4 
14  21 

29  30 
13  4 
3 
 5 
12  22 

2  
11  
25 

 ⇒ 
A41
1  6 
15  19 
24 
10  4 
14  21 
16 
29  30 
13  4 
3 
23  5 
12  22 
3 
2  20 
11  7 
25 

4  3 
...  1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 13 
30  29 
...  25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
This completes the 5x5 Magic Square Wheel Spoke Shift method. To see the Shift wheel type variant 5x5 square.
Go back to homepage.
Copyright © 2014 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com