NEW MAGIC SQUARES WHEEL METHOD
Part II
7x7 Wheel Border Magic Square
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).
A modified facile method for the construction of wheel type magic squares is now available. The position of the spokes are rotated by 90° so that
the left diagonal starts at the bottom left cell. The 7x7 as well as its internal internal squares.
In addition, eversion of the square gives an opposite square which is not bordered.
The new magic squares with n = 7 are constructed as follows using a complimentary table as a guide.
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 
A 7x7 Magic Square Using the Pairs {24,23,22,25,28,27,26} and {6,5,4,25,46,45,44}
 The 7x7 square is to be filled with 25 numbers from the subset 19 and their complements 4149 and the numbers 2228.
The spokes of the wheel are generated as follows: Numbers 2228 in the left diagonal; numbers 4,5,6 and conjugates 44,45,46
in the right diagonal; numbers 1,2,3 and conjugates 47,48,49 in top to bottom center; and 7,8,9 and conjugates 41,42,43 in center horizontal (square A1). The addition
of these pair of numbers and conjugates to the 7x7 square are shown below using directional pointed arrows:
1  2  3 
4  5  6 
7  8  9 
... 
22  23  24 
 25 
49  48  47 
46  45  44 
43  42  41 
... 
26  27  28 


↓  ↖ 
→  ...  ↗ 
 Sum up the rows and columns 13 and 57 and subtract from the magic sum 175. This gives the amounts required (shown in green Square A2) The last column shows the
two amounts need to complete the row and column (shown in yellow).
 Fill in the internal square 5x5 with the numbers 1013 and complements 3740 according to the picture below using two adjacent pair of numbers.

Then similarly fill in the external nonspoke cells (rows 1 and 7 and columns 1 and 7) with the numbers 14 to 21 and complements 29 to 36.
A1
44   
3   
26 
 45  
2   27 

  46  1 
28   
9  8  7 
25  43 
42  41 
  22 
49  4 
 
 23  
48  
5  
24   
47   
6 

⇒ 
A2
44   
3   
26  102  51+51 
 45  
2   27 
 101  50+51 
  46  1 
26   
100  50+50 
9  8  7 
25  43 
42  41 
  22 
49  4 
  100  50+50 
 23  
48  
5   99  49+50 
24   
47   
6  98  49+49 
98  99  100 
 100  101 
102   

⇒ 
A3
44   
3   
26 
 45  11 
2  40  27 

 37  46 
1  28 
13  
9  8  7 
25  43 
42  41 
 12  22 
49  4 
38  
 23  39 
48  10 
5  
24   
47   
6 

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

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

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 
Conversion of the 7x7 into its transposed opposite
Generation of a 7x7 transposed opposite can also follow the route used above. The method to transpose columns followed by rows follows.
This generates a new square in which only the external 7x7 is magic.
 Take square A4 and transpose (column 1 with column 3) and (column 5 with column 7) to get Square A5.
 Take square A6 and transpose (row 1 and row 3) and (row 5 with row 7) to get Square A6.
 In a sense A4 has been imploded or everted into A6, i.e., A4 and A6 below are opposites.
A4
44  15  17 
3  34  36 
26 
31  45  11 
2  40  27 
19 
29  37  46 
1  28 
13  21 
9  8  7 
25  43 
42  41 
20  12  22 
49  4 
38  30 
18  23  39 
48  10 
5  32 
24  35  33 
47  16  14 
6 

⇒ 
A5
17  15  44 
3  26  36 
34 
11  45  31 
2  19  27 
40 
46  37  29 
1  21 
13  28 
7  8  9 
25  41 
42  43 
22  12  20 
49  30 
38  4 
39  23  18 
48  32 
5  10 
3  35  24 
47  6  14 
16 

⇒ 
A6
46  37  29 
1  21  13 
28 
11  45  31 
2  19  27 
40 
17  15  44 
3  26 
36  34 
7  8  9 
25  41 
42  43 
3  35  24 
47  6 
14  16 
39  23  18 
48  32 
5  10 
22  12  20 
49  30  38 
4 

The result is a new everted square conforming to the same complementary table.
This completes part II of a border Magic Square Wheel method. To see the next 9x9 Part III.
Go back to homepage.
Copyright © 2015 by Eddie N Gutierrez.