NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part P3
How to Spoke Shift 9x9 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 methods used for the construction of border wheel type squares except that the initial spoke parts are added in a somewhat
different manner than in the original wheel method. The method consists of forming an internal 3x3 magic square, then generating all subsequent border magic squares.
as was done in the original method. The difference between this type of square and the original is that the left diagonal
numbers don't have to be chosen from the consecutive group ½(n^{2}n+2) to ½(n^{2}+n)
but may be chosen from any other consecutive group of numbers. However, the spoke of the central column is no longer the adjacent numbers
(37,38,39,40,41,42,43,44,45). The only numbers retained are (39,41,43) with the remaining three complements (40,42), (38,44) and (37,45) being replaced by another complementary
pair from the list:
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 
for the central column and not for the diagonal as was done for the regular wheel algorithm.
The symbol Δ will be used to specify a number added to or subtracted from the constants a, b or c to generate the first internal 3x3 magic square.
In this case Δ is, consequently, obtained from the first number of the set (34,35,36,37) used to generate the
left diagonal, 41 − 35 = 6.
In addition, both 4n + 1
and 4n + 3 squares may be filled with the entire complement set.
3x3 template
c+Δ  a 
b+Δ 
a+2Δ 
b  c 
bΔ 
c+2Δ  a+Δ 
A 9x9 Transposed Magic Square Using the Diagonals {30,31,32,33,41,49,50,51,52} and {38,37,36,35,41,47,46,45,44}

To the center column of the internal 3x3 square fill numbers ½(n^{2}1) to ½(n^{2}+3)
in consecutive order starting at the bottom middle cell of the 3x3 internal square and proceeding to the top middle cell of the 3x3 internal square
using the numbers listed in the complementary table described above, as
for example using n = 9. For a 9x9 square the numbers in the center column correspond to 39 → 41 → 43
starting from the 5^{th} row (Square A1).
 With 33, 35 and their complements generate a 3x3 square using Δ=6, b=41 and a=43 so that the sum of each column, row and diagonal of the 3x3 square
sums up to 123, the sum of the internal 3x3 square within a 9x9 square (Square A1).
 Generate Square A2 by adding consecutive numbers to the two diagonals and to the central column and row replace 37,38 and 40
with 28, 29 and 34, viz,
(the "spoke") numbers, and include their complements from the complement list above.
The reason for this connectivity will become apparent as we proceed. See Figure A where 37 and 38 are now on the left diagonal.
 To begin fill up the square add up the entries on the first row and subtract from 369 (the magic sum for a 9x9 square). This affords the value 241 which may be
give the sum of pairs needed to fill up that line, as 83 x 2 + 79. See Figure A2.
 Repeat for row 2 except subtract the value from 287 (the magic sum for a 7x7 internal square). This gives a value of 158 (65 + 93).
 Repeat for row 3 except subtract the value from 205 (the magic sum for a 5x5 internal square). This gives a value of 79.
 Do the same for rows 7, 8 and 9 obtaining, respectively, 85, 170 and 251.
 Then repeat for columns 1, 2 and 3 obtaining, respectively, 243, 162 and 81.
 Finally repeat for columns 7, 8 and 9 obtaining, respectively, 83, 166 and 249.
 Fill the 3^{rd} & 7^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(19,22) & (20,21) and enter into Square A3.
 Fill the 2^{nd} & 8^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(42,59),(12,1) & (10,11),(8,9) and enter into Square A4.
 Fill the 1^{st} & 9^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(15,17),(16,17),(13,14) & (6,7),(4,5),(2,3) and enter into Square A5.
 Figure A shows the connectivity between numbers in the complementary table where the red bars are the
"spoke" numbers. The same for their complements.
Figure A
 Square A6 shows the 4 border squares in "border format".
 The complement table below also shows how the color pairs are layed out (for comparison with Square A5).
A1
   
   

   
   

     
  
   33 
43  47 
  
  
55 
41  27 
  
   35 
39 
49    
   
 
  
   
  
 
   
   


⇒ 
A2
30    
54    
44  241 
 31   
53    45 
 158 
  32   48  
46    79 
   33 
43  47 
   
58  57  56 
55  41  27 
26  25  24 

   35 
39 
49     
  36  
34  
50    85 
 37   
29   
51   170 
38    
28    
52  251 
243  162  81 
  
83  166  249 


⇒ 
A3
30    
54    
44 
 31   
53    45 

  32  19 
48  60 
46   
  20  33 
43  47 
62   
58  57  56 
55  41  27 
26  25  24 
  61  35 
39 
49  21   
  36  63 
34  22 
50   
 37   
29   
51  
38    
28    
52 

 ⇒ 
A4
30    
54    
44 
 31  42 
12 
53  81 
23  45 

 10  32  19 
48  60 
46  72  
 8  20  33 
43  47 
62  74  
58  57  56 
55  41  27 
26  25  24 
 73  61  35 
39  49 
21  9  
 71  36  63 
34  22 
50  11  
 37  40 
70  29 
1  59 
51  
38    
28    
52 

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

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


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  27 
 
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  55 


28  29 
30  31  32 
33  34 
35  36  37 
38  39  40 
 41 

54  53 
52  51 
50  49 
48  47 
46  45  44 
43  42 
This completes the Part P3 of a 9x9 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part P4.
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Copyright © 2014 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com