NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part M6
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, every spoke on the wheel consists of consecutive numbers and their complements.
For example, for n = 11, we can choose the complementary numbers (56,57,58,59,60,61,62,63,64,65,66) from the complimentary table below:
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 (51,52,53,54,55) used to generate the left diagonal, 61 − 51.
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.
Figure A shows the connectivity of a 11x11 set. However, when Δ is an an odd number as for a 11x11 square at least one number in the set must be less than or
equal to 0. This is shown in for a 5x5 square Part M1 and a variant square 5x5 Part M2.
3x3 template
c+Δ  a 
b+Δ 
a+2Δ 
b  c 
bΔ 
c+2Δ  a+Δ 
A 11x11 Transposed Magic Square Using the Diagonals {46,47,48,49,50,61,72,73,74,75,76} and {55,54,53,52,51,61,71,70,69,68,67}
 Generate a 3x3 square using Δ=10, b=61 and a=62. (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 492 which divided by 4
gives the sum of pairs needed to fill up that line, which in this case is (123 x 4). 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 123x3.
 Repeat for row 3 except subtract the value from 427 (the magic sum for a 7x7 internal square). This gives a value of 123x2.
 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 8, 9 10 and 11 obtaining, respectively, 484, 363, 242 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,
(45,44) &(42,43) 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),(33,32) & (30,31),(28,29) 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,
(27,26),(25,24)(23,22) & (20,21),(18,19),(16,17) 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,
(15,14),(13,12),(11,10),(9,8) & (6,7),(4,5),(2,3),(0,1) and enter into Square A6.
 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
    
   
 
    
    

    
   
 
    
   
 
    50 
62  71 
   
  
 82 
61  40 
 
 
    51 
60  72 
   
    
   
 
    
   
 
    
    

    
   
 

⇒ 
A2
46     
66    
 67  492 
 47    
65     68 
 369 
  48   
64    69 
  246 
   49  
63   70  
  123 
    50 
62  71  
   
86 
85  84  83 
82  61  40 
39  38  37 
36  
    51 
60 
72      
   52  
59  
73     121 
  53   
58   
74    242 
 54    
57     75 
 363 
55     
56     
76  484 
484  363  242 
121   
 123 
246  369  492 


⇒ 
A3
46     
66    
 67 
 47    
65     68 

  48   
64    69 
 
   49  45 
63  78 
70    
   42  50 
62  71 
80    
86 
85  84  83 
82  61  40 
39  38  37 
36 
   79  51 
60  72 
43    
   52  77 
59  44 
73    
  53   
58   
74   
 54    
57     75 

55     
56     
76 

 ⇒ 
A4
46     
66    
 67 
 47    
65     68 

  48  35 
33  64  90 
88  69   
  30 
49  45 
63  78 
70  92   
  28  42  50 
62  71 
80  94   
86 
85  84  83 
82  61  40 
39  38  37 
36 
  93  79  51 
60 
72  43 
29   
  91  52 
77 
59  44 
73  31   
  53  87 
89  58 
32  34 
74   
 54    
57     75 

55     
56     
76 

⇒ 
A5
46     
66    
 67 
 47  27 
25  23 
65  100 
98 
96  68  
 20  48 
35  33 
64  90 
88  69 
102  
 18  30 
49  45 
63  78 
70  92 
104  
 16  28 
42  50 
62  71 
80  94  106  
86 
85  84  83 
82  61  40 
39  38  37 
36 
 105  93  79 
51  60 
72  43 
29  17  
 103  91 
52  77 
59  44 
73  31 
19  
 101 
53  87 
89  58 
32  34 
74  21  
 54  95 
97  99 
57  22  24 
26  75  
55     
56     
76 

⇒ 
A6
46  15 
13  11 
9  66 
114  112 
110  108  67 
6  47  27 
25  23 
65  100 
98 
96  68 
116 
4  20  48 
35  33 
64  90 
88  69 
102  118 
2  18 
30 
49  45 
63  78 
70  92 
104  120 
0  16  28 
42  50 
62  71 
80  94  106 
122 
86 
85  84  83 
82  61  40 
39  38  37 
36 
121  105  93 
79 
51  60 
72  43 
29  17  1 
119  103  91 
52  77 
59  44 
73  31 
19  3 
117  101 
53  87 
89  58 
32  34 
74  21  5 
115  54  95 
97  99 
57  22 
24 
26  75 
7 
55  107 
109  111 
113  56 
8  10 
12  14 
76 

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