NONCONSECUTIVE MAGIC SQUARES WHEEL METHOD (Part V)
A Discussion of the nonconsecutive Magic Square Wheel Method
Construction of these magic squares requires is an approach which differs from the traditional step methods such as the Loubère and Méziriac.
The magic square is constructed as follows using a complimentary table as a guide. The cells in the same color are associated with one another; for example,
only 3 → 4 → 5 → 45 → 46 → 47 are still pairassociated consecutively together, while the other colored pairs are not.
Because of the way these pairs are grouped the 7x7 square produced will not be initially magic but must be converted into one by a series of steps. In addition, on this
page we must perform a double modification to arrive at a magic partially nonconsecutive square.
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 

 The left diagonal is filled with the group of numbers ½
(n^{2}n+2) to ½(n^{2}+n) in consecutive
order (top left corner to the right lower corner) from the numbers listed in the complementary table described above, for example using n = 7.
For a 7x7 square the numbers in the left diagonal correspond to 22 → 23 → 24 → 25 → 26 → 27 → 28 (Square A1).
Then add the right diagonal in reverse order from bottom left corner to the right upper corner choosing only from the pair {4,5,7,43,45,46}.
 This is followed by the central column from the pairs {2,3,6,44,47,48} in regular order and
then by the central row from the pairs {1,8,9,41,42,49} in reverse order (Square A2). We now have a partial square with not all sums equal to 175.
 The result of these operations figuratively speaking resembles the hub and spokes of a wheel where the cells in color correspond to the spoke and hub of the
wheel, with the nonconsecutive pairs in different colors.
A1
22  
 

 46 
 23 
 

45  
 
24  
43 
 
 
 25 
 

 
7  
26   
 5 
 
 27  
4  
 
  28 

⇒ 
A2
22  
 2 
  46 
 23 
 3 

45  
 
24  6 
43 
 
49  42 
41  25 
9  8 
1 
 
7  44 
26   
 5 
 47 
 27  
4  
 48 
  28 

⇒ 
 Fill in the spoke portions of the square with the twelve pairs that are left over (Square A3 and A4). This is done as follows:
Use the small squares (from the complementary table) as an aid in in how to place the adjacent pair of numbers (in white),
which may be placed in either a clockwise or anticlockwise manner as shown below:
10   40 

10   40 
↓  ↗  ↓ 

↑  ↙  ↑ 
40   39 

40   39 
A3
 175 
22  10 
12  2 
38  40  46  170 
14  23 
 3 

45  35  121 
16  
24  6 
43 
 33  122 
49  42 
41  25 
9  8 
1  175 
34  
7  44 
26   17  128 
36  5 
 47 
 27  15  130 
4  39 
37  48 
13  11  28  180 
175  119  121 
175  129  131 
175  175 

⇒ 
A4
 175 
22  10 
12  2 
38  40  46  170 
14  23 
18  3 
32 
45  35  170 
16  20 
24  6 
43 
29  33  171 
49  42 
41  25 
9  8 
1  175 
34  30 
7  44 
26  21  17  179 
36  5 
31  47 
19  27  15  180 
4  39 
37  48 
13  11  28  180 
175  169  170 
175  180  181 
175  175 

⇒ 
 At this point all sums are not equal to 175 so modifications are carried out to generate a magic square. Convert {2,3,6,44,47,48}, respectively, to {7,8,10,40,42,43}.
This converts all sums in the last grey row to 175 (Square A5).
 Also do this to the center row by converting {42,41,9,8} to {48,46,4,2}, respectively, which converts all sums in the last row to 175 (Square A6).
A5
 175 
22  10 
12  7 
38  40  46  175 
14  23 
18  8 
32 
45  35  175 
16  20 
24  10 
43 
29  33  175 
49  42 
41  25 
9  8 
1  175 
34  30 
7  40 
26  21  17  175 
36  5 
31  42 
19  27  15  175 
4  39 
37  43 
13  11  28  175 
175  169  170 
175  180  181 
175  175 

⇒ 
A6
 175 
22  10 
12  7 
38  40  46  175 
14  23 
18  8 
32 
45  35  175 
16  20 
24  10 
43 
29  33  175 
49  48 
46  25 
4  2 
1  175 
34  30 
7  40 
26  21  17  175 
36  5 
31  42 
19  27  15  175 
4  39 
37  43 
13  11  28  175 
175  175  175 
175  175  175 
175  175 

 At this point 6 duplicates are present {4,7,10,40,43,46}. Four can be removed first followed by the last two. For example, by adding
n^{2} = 49 to each of {4,7,10,16,27,35,43} square A7 is obtained.
 Adding n^{2} = 49 to each of {14,20,37,40,46,53,76} removes the other two duplicates to give Square A8 with the last modifications
in yellow where S = 273 and S = ½(n^{3} + 29n).
 The last equation takes into account that the sum has been modified from the known equation
S = ½(n^{3} + n) to
the general equation as was shown in:
S = ½(n^{3} ± an)
which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or 
values. For example when a = 1 the normal magic sum S is implied.
When a takes on different odd values S gives the magic sum of a modified magic square.
It can be seen from the equation that addition or subtraction of n^{2} to some of the cells in
the square gives rise to a new magic square, which in this case have cell numbers greater than 49.
>
A7
 224 
22  59 
12  7 
38  40  46  224 
14  23 
18  8 
32 
45  84  224 
65  20 
24  10 
43 
29  33  224 
49  48 
46  25 
53  2 
1  224 
34  30 
56  40 
26  21  17  224 
36  5 
31  42 
19  76  15  224 
4  39 
37  92 
13  11  28  224 
224  224  224 
224  224  224 
224  224 

⇒ 
A8
 273 
22  59 
12  7 
38  40  95  273 
63  23 
18  8 
32 
45  84  273 
65  69 
24  10 
43 
29  33  273 
49  48 
46  25 
102  2 
1  273 
34  30 
56  89 
26  21  17  273 
36  5 
31  42 
19  125  15  273 
4  39 
86  92 
13  11  28  273 
273  273  273 
273  273  273 
273  273 

 In addition, the modified complementary table is depicted below with the modified cell numbers in yellow
and green where some of the numbers no longer show complementarity. Also note that 7, 10 40 and 46, not shown on the
complementary table because of duplication, are present in Square A8.
1  2 
3  4 
5  6 
56  8 
102  59 
11  12 
13  63 
15  65 
17  18 
19  69 
21  22 
23  24 

 25 
49  48 
47  95 
45  44 
92  42 
41  89 
39  38 
86  36 
84  34 
33  32 
31  30 
29  28 
125  26 

This completes Part V of the nonconsecutive Magic Square Wheel method.
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Copyright © 2009 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com