NEW MAGIC SQUARE METHODS: WHEEL EXPANSION
A Discussion of new Method B Algorithm
To construct a magic square by this method the series 1..n^{2}
is paired up in complementary fashion, for example 1 is paired with
n^{2}, 2 with n^{2}1, etc. These pair of numbers
are eventually partitioned into n+1 groups of ½(n1) pairs
including the unpaired middle number of the series. However as opposed to Method A, this partioning is done by choosing triplets of pairs
(the numbers and their complements) for the "spoke" numbers and doublets of pairs for the
"nonspoke" numbers
and filling the entire wheel from the inside out, as shown below.
In other words, a "spoke" is generated from a triplet of pairs for a 3x3 square or a multiple of triplets
{a,complement a),(b,complement b),(c,complement c)}... where the equation n  ½(n + 1) gives the number of triplets
per nxn square along with the hub.
For example, for a 5x5 square II (below), the two triplets {1,2,3} and {4,5,6} along with their complements {25,24,23} and {22,21,20} are taken from
the complementary table below and used to construct the "spokes".
By analysis the number of conformations is
equal to the equation {(Σ ¼(n^{2}4n + 7)  a} × (½(n1))!
where a is varied from 0 to n1. This method differs from method A in that each group of three pairs may take any available
concentric positions on the square.
On the other hand, the use of the equation
{ ¼(n^{2}4n + 7) × ½ (¼(n^{2}4n + 7) + 1) × (½
(n1))!}, avoids having to use summation groups.
For a 5x5 square, however, either of these equations generates 12 conformations.
Method B
A magic square is first constructed by filling in the left diagonal with the group of numbers ½
(n^{2}n+2) to ½(n^{2}+n) using any of the requisite combinations. For a 5x5 square
the numbers in the left diagonal are added consecutively for example using variant 1 {11,12,13,14,15} as in square I.
The central square is first laid down according to the wheel algorithm, followed by the external second square (as in Square II).
The "nonspoke" are finally added according to the parity table on the right.
Square I
11  
 

 12 
3  24 

 25 
13  1 

 2 
23  14 

 
 
15 

⇒ 
Square II
11  
6  
21 
 12 
3  24 

22  25 
13  1 
4 
 2 
23  14 

5  
20  
15 


ROWS  PAIR  PARITY 
1  27  O 
2  26  E 
4  26  E 
5  25  O 

1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 
1  2 
3  4 
5  6 
7  8 
9  10 
11  12 

 13  
 13 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 

The result is Square III and IV, where both square IV and its concentric square are also magic.
Square III
11  19 
6  8 
21 
 12 
3  24 

22  25 
13  1 
4 
 2 
23  14 

5  7 
20  18 
15 

⇒ 
Square IV
11  19 
6  8 
21 
10  12 
3  24 
16 
22  25 
13  1 
4 
17  2 
23  14 
9 
5  7 
20  18 
15 

Alternatively, the "spoke" may be formed by using different combination of numbers.
For example the square produced by using {4,5,6} followed by {1,2,3}
gives square V then VI which may then be filled in the usual fashion.
Square V
11  
 

 12 
6  21 

 22 
13  4 

 5 
20  14 

 
 
15 

⇒ 
Square VI
11  
3  
24 
 12 
6  21 

25  22 
13  4 
1 
 5 
20  14 

2  
23  
15 

1  2 
3  4 
5  6 
7  8 
9  10 
11  12 
 
1  2 
3  4 
5  6 
7  8 
9  10 
11  12 

 13  
 13 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 
 
25  24 
23  22 
21  20 
19  18 
17  16 
15  14 

One example of a 7x7 spinning magic square
One example (out of 168) of a 7x7 square is the combination
24 → 22 → 27 → 25 → 23 → 28 → 26.
The first concentric wheel is produced using for example the first triplet (13) along with the last three triplets (2224)
and the central number (25) as shown in square VII. The rest of the "spoke" numbers are added
so that the sum of three numbers per row is closest to 75 (the magic sum of a central 7x7 square).
Thus between the numbers 1 and 3 the best choice is 1 since the sum produced is 73 as opposed to 71 with the number 3.
Repeating for rows one and two using this same method gives square VIII. The "nonspoke" numbers
are then filled in according to the following parity table,
remembering that the allowable pair sums are n^{2} = 49, n^{2}+1 = 50 and n^{2}+2 = 51 where these
are compared to the possible nonallowed alternatives (note that only half of the parity table is shown since by symmetry the rest of the table is similar):
Row/Column  Number  Pair of Sums  Parity  Allowed 

1  9  50+50  E+E  Yes 
1  7  51+51  O+O  No 
2  6  51+51  O+O  Yes 
2  4  52+52  E+E  No 
3  3  47+52  O+E  No 
3  3  48+51  O+E  No 
3  1  49+50  O+E  Yes 
For example, although line 2 of the parity table contains an allowed O+O, no allowed E+E is available for lines 47. Therefore, the allowed line is 1.
Square VII
24  
 
 

 22 
 
 

 
27  1 
48  

 
47  25 
3  

 
2  49 
23  

 
 
 28 

 
 
 
26 

⇒ 
Square VIII
24  
 9 
 
42 
 22 
 6 
 45 

 
27  1 
48  

43  46 
47  25 
3  4 
7 
 
2  49 
23  

 5 
 44 
 28 

8  
 41 
 
26 

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

We first try filling row three with "nonspoke" numbers. However, placing the sum of allowable pairs 50 and 49
is not as easy as it seems. The first thing to do is to fill in row/column 2 (as shown in square IX) since both pair of numbers are 51. This makes it
easier to fill in the rest of the square
since row/column three must be filled in with the pair that adds up to 49 (as shown in square X). Note that the number inserted is 31 a reverse H.
Square IX
24  36 
 9 
 14 
42 
40  22 
38  6 
13  45 
11 
 34 
27  1 
48  16 

43  46 
47  25 
3  4 
7 
 17 
2  49 
23  33 

10  5 
12  44 
37  28 
39 
8  15 
 41 
32  
26 

⇒ 
Square X
24  36 
31  9 
19  14 
42 
40  22 
38  6 
13  45 
11 
29  34 
27  1 
48  16 
20 
43  46 
47  25 
3  4 
7 
21  17 
2  49 
23  33 
30 
10  5 
12  44 
37  28 
39 
8  15 
18  41 
32  35 
26 

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

This completes method B. To continue on to brand new Loubère and wheel methods.
To go back to the previous Method A2 or to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com