NEW MAGIC SQUARES WHEEL METHOD
Part III
9x9 Magic Square Wheel
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 9x9 square constructed has no internal border squares.
In addition the partially bordered square may be everted to give an opposite square whose internal 7x7 square is not magic but whose 3x3, 5x5 and 9x9 are magic.
The new magic squares with n = 9 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 
 
81  80  79 
78  77  76  75 
74  73  72  71 
70  69  68  67 
66  65  64  63 
62 

21  22 
23  24  25  26  27 
28  29  30  31  32 
33  34  35  36  37 
38  39  40 
 41 
61  60  59 
58  57  56  55 
54  53  52  51 
50  49  48  47 
46  45  44  43 
42 
A 9x9 Transposed Magic Square Using the Diagonals {37,38,39,40,41,42,43,44,45} and {5,6,7,8,41,74,75,76,77}
 The 9x9 square is to be filled with 33 numbers from the subset 112 and their complements 7081 and the numbers 3745.
The spokes of the wheel are generated as follows: Numbers 3745 in the left diagonal; numbers 5,6,7,8 and conjugates 74,75,76,77
in the right diagonal; numbers 1,2,3,4 and conjugates 76,79,80,81 in top to bottom center; and 9,10,11,12 and conjugates 70,71,72,73 in center horizontal (square A1).
The addition of these pair of numbers and conjugates to the 9x9 square are shown below using directional pointed arrows:
1  2  3 
4  5  6 
7  8  9 
10  11  12 
... 
37  38  39 
40 
 41 
81  80  79 
78  77  76 
75  74  73 
72  71  70 
... 
45  44  43 
42 


↓  ↖ 
→  ...  ↗ 
 Sum up the rows and columns 14 and 69 and subtract from the magic sum 369. 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 nonspoke cells of the internal 5x5 square with the numbers 21 to 24 and complements 58 to 61 as shown in Square A3 using four adjacent pair of
numbers according to inset C in the picture:
 Fill in the nonspoke cells of the outer rows of the internal square 7x7 with the numbers 1324 and complements 6269 according to inset A or B above using
two adjacent pair of numbers.
 Fill in the nonspoke cells of the outer rows of the external square 9x9 with the numbers 2536 and complements 4657 according to inset A above using two
adjacent pair of numbers.
 Note that the nonspoke numbers of the entire 7x7 square is laid out according to the method of Part I and is,
therefore, easy to construct. The outer rows and columns of the 9x9 square then just require either two numbers summing up to 81 or 83.
A1
77    
1    
45 
 76   
2    44 

  75   3  
43   
   74 
4  42 
  
9  10  11 
12 
41  70 
71  72  73 
   40 
78 
8    
  39  
79  
7   
 38   
80   
6  
37    
81    
5 

⇒ 
A2
77    
1    
45  246  82x3 
 76   
2    44 
 247  83x2+81 
  75   3  
43    248 
83x2+82 
   74 
4  42 
   249  82x2+84 
9  10  11 
12 
41  70 
71  72  73 
 
   40 
78  8   
 243  82x2+80 
  39  
79   7 
  244  81x2+82 
 38   
80    6 
 245  81x2+83 
37    
81    
5  246  82x3 
246  245  244 
243   249 
248  247  246 
 

⇒ 
A3
77    
1    
45 
 76   
2    44 

  75  22 
3  61 
43   
  60  74 
4  42 
24   
9  10  11 
12 
41  70 
71  72  73 
  21  40 
78 
8  59   
  39  58 
79  23 
7   
 38   
80   
6  
37    
81    
5 

 ⇒ 
A4
77    
1    
45 
 76  13 
16 
2  67 
68  44 

 64  75  22 
3  61 
43  18  
 63  60  74 
4  42 
24  19  
9  10  11 
12 
41  70 
71  72  73 
 20  21  40 
78  8 
59  62  
 17  39  58 
79  23 
7  65  
 38  69 
66  80 
15  14 
6  
37    
81    
5 

⇒ 
A5
77  25 
27  29 
1  53  55 
57  45 
51  76  13 
16  2  67 
68  44  32 
49  64 
75  22 
3  61 
43  18  34 
47  63 
60  74 
4  42 
24  19  35 
9  10  11 
12 
41  70 
71  72  73 
36  20  21 
40  78 
8  59 
62  46 
33  17 
39  58 
79  23 
7  65  48 
31  38  69 
66  80 
15  14 
6  50 
37  56 
54  52 
81  30  28 
26  5 
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 
26 
 
81 
80  79 
78  77 
76  75 
74  73 
72  71 
70  69 
68  67 
66  65 
64  63 
62  61 
60  59 
58  57 
56 

 27 
28  29 
30  31  32 
33  34 
35  36  37 
38  39  40 
 41 
 55 
54  53 
52  51 
50  49 
48  47 
46  45  44 
43  42 
Conversion of the 9x9 into its transposed opposite
Generation of a 9x9 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 4), (column 2 with column 3), (column 6 with column 9) and (column 7 with column 8) to get Square A6.
 Take square A6 and transpose (row 1 with row 4), (row 2 with row 3), (row 6 with row 9) and (row 7 with row 8) to get Square A7.
 In a sense A5 has been imploded or everted into A7, i.e., A5 and A7 below are opposites.
A5
77  25 
27  29 
1  53  55 
57  45 
51  76  13 
16  2  67 
68  44  32 
49  64 
75  22 
3  61 
43  18  34 
47  63 
60  74 
4  42 
24  19  35 
9  10  11 
12 
41  70 
71  72  73 
36  20  21 
40  78 
8  59 
62  46 
33  17 
39  58 
79  23 
7  65  48 
31  38  69 
66  80 
15  14 
6  50 
37  56 
54  52 
81  30  28 
26  5 

⇒ 
A6
29  27 
25  77 
1  45  57 
55  53 
16  13  76 
51 
2  32 
44  68 
67 
22  75 
64  49 
3  34 
18  43  61 
74  60 
63  47 
4  35 
19  24  42 
12  11  10 
9 
41  73 
72  71  70 
40  21  20 
36  78 
46  62 
59  8 
58  39 
17  33 
79  48 
65  7  23 
66  69  38 
31  80 
50  6 
14  15 
52  54 
56  37 
81  5 
26  28 
30 

⇒ 
A7
74  60 
63  47 
4  35 
19  24  42 
22  75 
64  49 
3  34 
18  43  61 
16  13  76 
51  2  32 
44  68  67 
29  27 
25  77 
1  45  57 
55  53 
12  11  10 
9  41  73 
72  71  70 
52  54 
56  37 
81  5 
26  28 
30 
66  69  38 
31  80 
50  6 
14  15 
58  39 
17  33 
79  48 
65  7  23 
40  21  20 
36  78 
46  62 
59  8 

⇒ 
A7 Partially Bordered
74  60 
63  47 
4  35 
19  24  42 
22  75 
64  49 
3  34 
18  43  61 
16  13  76 
51  2  32 
44  68  67 
29  27 
25  77 
1  45  57 
55  53 
12  11  10 
9  41  73 
72  71  70 
52  54 
56  37 
81  5 
26  28 
30 
66  69  38 
31  80 
50  6 
14  15 
58  39 
17  33 
79  48 
65  7  23 
40  21  20 
36  78 
46  62 
59  8 

The result is a new square conforming to the same complementary table above and where the 3x3, 5x5 and 9x9 are magic but the 7x7 border square is not.
This completes Part III of a 9x9 border Magic Square Wheel method. To go to Part IV of an 9x9 square.
Go back to homepage.
Copyright © 2015 by Eddie N Gutierrez