NEW MAGIC SQUARES WHEEL METHOD  BORDER SQUARES
Part X1
How to generate 9x9 Border 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 border wheel type squares. The method consists of forming a 5x5 internal Loubère
magic square then filling in the external 1,2 and 8,9 rows and columns with the requisite nonLoubère numbers as will be shown below.
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 
Furthermore, the symbol δ (where δ = 4) specifies the difference between entries on the diagonals and center row and column not situated on the
center 5x5 square (shown in square A1)
The nonLoubère entries in rows and columns 1,2 and 8,9 are added according to a coded system ( which I call "coded connectivity"
as opposed to lined connectivity) employing a number and superscript and where the number gives the difference between two paired numbers and the superscript shows which
two numbers are paired together.
For example, 11^{1} says that this number is added to a second complementary number 11^{1} separated by a distance of 11. From the complementary table above
1 + 71 is such an example.
While, 7^{a} means that this number is added to a noncomplementary number 7^{a} both which are 7 units apart. In addition, if we look at the
complementary table above 2^{1} corresponds to the sum of 1 + 80, while 2^{1} to the sum of 2 + 81.
When either of the two sums is required, the number is preceded by either a ( ) or by ().
A 9x9 Transposed Magic Square Using the Diagonals {60,56,45,33,41,49,37,26,22} and {24,28,39,40,41,42,43,54,58}
 Fill in the 5x5 Loubère internal square with the numbers 2953 (square A1).
 Subtract δ = 4 from 29 and add this number (25) to the center cell of row 2 and repeat again (254) = 21 and place this number in the center cell of row 1.
Starting with the number 21 place consecutive numbers as shown in Square A1 in a spiraling fashion up to the number 28, followed by their complementary
numbers (Square A1). This completes the spokes for the square, along with the nonspoke numbers of the 5x5 square.
 Sum up the empty 1^{st} row, the empty 9^{th};
the 2^{nd}, the 8^{th} rows.
Do the same for the columns (green cells). See Square A2.
 Fill in the internal 7x7 square with numbers generated using the new coding method (Square 3). For example in row 2 using numeric superscripts, 2 is added to 74
and 4 to 72 , followed by their complements. While in column 2 also using numeric superscripts, 11 is added to 77 and 18 to 70 also followed by their complements.
 Finally fill in the external 9x9 square (color cells) (Square A4). For example, in row 1, 3 is added to 75, 13 to 65 and 1 to 73.
While in column 9, 67 is added to 19, 66 to 20 and 76 to 14 followed by their complements, using numeric superscripts.
 Below is the coded connections to this square where the colored "spoke" cells are not included in the coding:
1  2  3  4  5  6  7  8  9 
10  11  12  13  14  15  16  17  18  19 
20  ...  40 
 41 
81  80  79  78  77  76  75  74  73 
72  71  70  69  68  67  66  65  64  63 
62  ...  42 


9^{1}  7^{1}  5^{1}  7^{2} 
7^{3}  9^{2}  5^{1}  7^{1}  9^{1}  7^{2} 
7^{3}  7^{4}  5^{2}  9^{2}  5^{3} 
5^{4}  5^{2}  7^{4}  5^{3}  5^{4}  ... 
 Because of the messy connectivities using spaghetti lines its best to use the connectivity table and the table at the end of this page to
assertain the connectivities.
 Square A5 shows the 3 border squares in "border format".
 The complementary table below also shows how the color pairs are layed out (for comparison with Square A4).
A1
60    
21    
58 
 56   
25    54 

  45  52 
29  36 
43   
  51  33 
35  42 
44   
23  27  32 
34  41  48 
50  55  59 
  38  40 
47 
49  31   
  39  46 
53  30 
37   
 28   
57   
26  
24    
61    
22 

⇒ 
A2 (δ=4)
60    
21    
58  230  78x2+74 
 56   
25    54 
 152  76x2 
  45  52 
29  36 
43    

  51  33 
35  42 
44    

23  27  32 
34  41  48 
50  55  59 
 
  38  40 
47 
49  31   
 
  39  46 
53  30 
37    

 28   
57   
26  
176  88x2 
24    
61    
22  262  2x86+90 
262  176  
  
 152  230 
 

⇒ 
A3
60    
21    
58 
 56  2 
4  25 
72  74 
54  
 77  45  52 
29  36 
43  5  
 70  51  33 
35  42 
44  12  
23  27  32 
34  41  48 
50  55  59 
 18  38  40 
47  49 
31  64  
 11  39  46 
53  30 
37  71  
 28  80 
78  57 
10  8 
26  
24    
61    
22 

⇒ 
A4
60  3 
13  1 
21  73  65 
75  58 
67  56  2 
4  25  72 
74  54  15 
66  77 
45  52 
29  36 
43  5  16 
76  70 
51  33 
35  42 
44  12  6 
23  27  32 
34  41  48 
50  55  59 
14  18  38 
40  47 
49  31 
64  68 
20  11 
39  46 
53  30 
37  71  62 
19  28  80 
78  57 
10  8 
26  63 
24  79 
69  81 
61  9  17 
7  22 

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

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 
This completes Part X1 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part X2 of an 9x9 square.
Go back to homepage.
Copyright © 2015 by Eddie N Gutierrez