NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part N9
How to Spoke Shift 11x11 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
(56,57,58,59,60,61,62,63,64,65,66). At least one pair of complements must be retained in the central column which may be
(57,58,59,60,61,62,63,64,65) for the replacement of one, (58,59,60,61,62,63,64) for the replacement of two, (59,60,61,62,63) for the replacement of three and
(60,61,62) for the replacement of sets of complementary pairs from the list:
1  2 
3  4  5  6 
7  8  9  10 
11  12  13  14 
15  16  17  18 
19  20 
 
121  120  119 
118  117  116  115 
114  113  112  111 
110  109  108  107 
106  105  104  103 
102 

21  22 
23  24  25  26  27 
28  29  30  31  32 
33  34  35  36  37 
38  39  40 

101  100  99 
98  97  96  95 
94  93  92  91 
90  89  48  87 
86  85  84  83 
82 

41  42 
43  44  45  46  47 
48  49  50  51  52 
53  54  55  56  57 
58  59  60 
 61 
81  80  79 
78  77  76  75 
74  73  72  71 
60  69  68  67 
66  65  64  63 
62 
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 (54,55,56,57,58) used to generate the left diagonal,
61 − 54 = 7.
In addition, 4n + 1 number behave differently from
4n + 3
in that in the former at least one number in the set must be less than or equal to 0, while in the latter all numbers from 1 to
½(n^{2} − 1) are usable.
3x3 template
c+Δ  a 
b+Δ 
a+2Δ 
b  c 
bΔ 
c+2Δ  a+Δ 
A 11x11 Transposed Magic Square Using the Diagonals {49,50,51,52,53,61,69,70,71,72,73} and {58,57,56,55,54,61,68,67,66,65,64}

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 = 11. For a 11x11 square the numbers in the center column correspond to 60 → 61 → 62
starting from the 5^{th} row (Square A1).
 With 52, 53 and their complements generate a 3x3 square using Δ=7, b=61 and a=62 so that the sum of each column, row and diagonal of the 3x3 square
sums up to 183, the sum of the internal 3x3 square within a 11x11 square (Square A1).
 Generate Square A2 by adding consecutive numbers to the two diagonals and the central column and row
(the "spoke") and include their complements from the complement list above.
 To begin fill up the square add up the entries on the first row and subtract from 671 (the magic sum for a 11x11 square). This affords the value 477 which although
not divisible by 4 requires three numbers that are equal and a fourth nonequal number to fill up that line, which in for example might be (121 x 3 + 114).
See Figure A2.
 Repeat for row 2 except subtract the value from 549 (the magic sum for a 9x9 internal square). This gives a value of 359 which equals 123 x 2 + 113.
 Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 123 x 2.
 Repeat for row 4 except subtract the value from 305 (the magic sum for a 5x5 internal square). This gives a value of 123.
 Do the same for rows 11, 10, 9 and 8 obtaining, respectively, 499, 373, 252 and 121.
 Then repeat for columns 1, 2, 3 and 4 obtaining, respectively, 484, 363, 242 and 121.
 Finally repeat for columns 11, 10, 9 and 8 obtaining, respectively, 492, 369, 246 and 123.
 Fill the 4^{th} & 8^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(40,39) &(37,38) and enter into Square A3.
 Fill the 3^{rd} & 9^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(35,34),(27,36) & (32,33),(30,31) and enter into Square A4.
 Fill the 2^{nd} & 10^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(29,28),(25,24)(27,26) & (22,23),(20,21),(18,19) and enter into Square A5.
 And finally fill the 1^{st} & 11^{th} rows & columns with the pairs/complements from the complement list corresponding to the requisite sums,
(8,16),(14,15),(12,13),(10,11) & (6,7),(4,5),(2,3),(0,1) and enter into Square A6.
 In addition, (9,113) cannot be included into the magicsquare since n is of type 4n + 3.
 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 A6).
A1
    
   
 
    
    

    
   
 
    
   
 
    53 
62  68 
   
  
 76 
61  46 
 
 
    54 
60  69 
   
    
   
 
    
   
 
    
    

    
   
 

⇒ 
A2
49     
81    
 64  477 
 50    
75     65 
 359 
  51   
74    66 
  236 
   52  
63   67  
  123 
    53 
62  68  
   
80 
79  78  77 
76  61  46 
45  44  43 
42  
    54 
60 
69      
   55  
59  
70     121 
  56   
48   
71    252 
 57    
47     72 
 373 
58     
41     
73  499 
484  363  242 
121   
 123 
246  369  492 


⇒ 
A3
49     
81    
 64 
 50    
75     65 

  51   
74    66 
 
   52  40 
63  83  67  
 
   37  53 
62  68  85 
  
80 
79  78  77 
76  61  46 
45  44  43 
42 
   84  54 
60 
69  38    
   55  82 
59  39 
70    
  56   
48   
71   
 57    
47     72 

58     
41     
73 

 ⇒ 
A4
49     
81    
 64 
 50    
75     65 

  51  35 
27  74  86 
88  66   
  32 
52  40 
63  83 
67  90   
  30  37  53 
62  68 
85  92   
80 
79  78  77 
76  61  46 
45  44  43 
42 
  91  84  54 
60 
69  38 
31   
  89  55 
82 
59  39 
70  33   
  56  87 
95  48 
36  34 
71   
 57    
47     72 

58     
41     
73 

⇒ 
A5
49     
81    
 64 
 50  29 
25  17 
75  96 
98 
94  65  
 22  51 
35  27 
74  86 
88  66 
100  
 20  32 
52  40 
63  83 
67  90 
102  
 18  30 
37  53 
62  68 
85  92  104  
80 
79  78  77 
76  61  46 
45  44  43 
42 
 103  91  84 
54  60 
69  38 
31  19  
 101  89 
55  82 
59  39 
70  33 
21  
 99 
56  87 
95  48 
36  34 
71  23  
 57  93 
97  105 
47  26  24 
28  72  
58     
41     
73 

⇒ 
A6
49  8 
14  12 
10  81 
111  109 
107  106 
64 
6  50  29 
25  17 
75  96 
98  94 
65  116 
4  22  51 
35  27 
74  86 
88  66 
100  118 
2  20 
32 
52  40 
63  83 
67  90 
102  120 
0  18  30 
37  53 
62  68 
85  92  104 
122 
80 
79  78  77 
76  61  46 
45  44  43 
42 
121  103  91 
84 
54  60 
69  38 
31  19  1 
119  101  89 
55  82 
59  39 
70  33 
21  3 
117  99 
56  87 
95  48 
36  34 
71  23  5 
115  57  93 
97  105 
47  26 
24  28 
72  7 
58  114 
108  110 
112  41 
11  13 
15  16 
73 

⇒ 
A7
49  8 
14 
12  10 
81  111 
109  107 
106  64 
6  50 
29 
25  17 
75  96 
98 
94  65 
116 
4  22 
51 
35  27 
74  86 
88  66 
100  118 
2  20 
32 
52  40 
63  83 
67  90 
102  120 
0  18 
30  37 
53  62  68 
85  92  104 
122 
80  79 
78  77 
76  61  46 
45  44  43 
42 
121  103 
91  84 
54  60  69 
38  31  19 
1 
119  101 
89  55 
82  59 
39 
70  33 
21  3 
117  99 
56  87 
95  48 
36  34 
71  23 
5 
115  57 
93 
97  105 
47  26 
24  28 
72  7 
58  114 
108  110 
112  41 
11  13  15 
16  73 
0  1  2 
3  4 
5  6 
7  8  9 
10  11 
12  13  14 
15  16 
17  18 
19 
 
122  121 
120  119 
118  117 
116  115 
114  113 
112  111 
110  109 
108  107 
106  105 
104  103 
20  21  22 
23  24  25 
26  27 
28  29 
30  31  32 
33  34 
35  36  37 
38  39  40 
41  42  43 
44  45 
 
102  101 
100  99 
98  97 
96  95 
94  93 
92  91 
90  89 
88  87 
86  85  84 
83  82 
81  80  79 
78  77 
46 
47  48  49 
50  51 
52  53  54 
55  56  57 
58  59  60 
 61 
76 
75  74  73 
72  71 
70  69  68 
67  66  65 
64  63  62 
This completes the Part N9 of a 11x11 Magic Square Wheel Spoke Shift method. To go forward to 11x11 Part N10.
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Copyright © 2014 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com