NEW MAGIC SQUARES WHEEL METHOD  SPOKE SHIFT
Part W5
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 numbers (and complements)
are added consecutively, starting from 1, at the center top cell. Subsequent numbers are added to each of the diagonals and the center row. The left diagonal of the
internal 3x4 square, however, deviates from this arrangement where the three numbers on this diagonal are
½(n^{2} − 1), ½(n^{2} + 1),
½(n^{2} + 3).
In addition, the symbol Δ which has been used to specify a number added to or subtracted from the constants a, b or c in the first
internal 3x3 magic square will always equal 1.
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 
Furthermore, a new symbol δ specifies the difference between entries on the diagonals and center row and column where
δ = 4 in all our cases except for the left diagonal of the internal 3x3 square (as shown below):
3x3 template
c+Δ  a 
b+Δ 
a+2Δ 
b  c 
bΔ 
c+2Δ  a+Δ 
To avoid spaghetti type connections between paired nonspoke numbers, a coded system ( which I call "coded connectivity" as opposed
to lined connectivity)
employs a number and superscript 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 such as adding 21 to the
complement of 31, i.e., 21 + 91 to give 112.
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 + 120, while 2^{1} to the sum of 2 + 121.
When either of the two sums is required, the ( ) or the () shows which one is being used.
This page unlike a previous method employs an internal 3x3 square containing the consecutive numerals 1, 2 and 3.
In addition, the numerals for the 5x5, 7x7 and 9x9 squares are incrementally added starting with the number 5 and increasing to the number 20.
See Square A1 below. The rest of the square is filled out as follows:
A 11x11 Transposed Magic Square Using the Diagonals {104,108,112,116,120,61,2,6,10,14,18} and {20,16,12,8,60,61,62,114,110,106,102}
 Add one to the first row center of a 11x11 square, 2 to the rightmost bottom cell, 3 to the center of the first column and 4 to the leftmost bottom cell.
Repeat (i.e. spiraling outwards from the center) starting with the number 5 up to the number 20, followed by their complementary numbers (Square A1).
 Add the numbers 60 and 62 to the empty two internal cells. This generates a 3x3 internal magic square (Square A1).
 Sum up the empty 1^{st} row, the empty 11^{th};
the 3^{rd}, the 10^{th}; the 3^{nd};the 9^{th} and the 4^{nd};the 8^{th } rows.
Do the same for the columns (green cells). The values are in the twelfth column and are equal to the multiplied values in the thirteenth column.
That is including both colums and rows, there should be 8 sums whose values are listed in column 12 (Square A2).
 Fill in the internal 5x5 square (green cells) with numbers generated using the new coding method (Square A3). For example in row 4, 4 is added to 66 using
numeric superscripts while in column 8, 33 is added to 37, followed by their complements, using alphabetic superscripts.
 Fill in similarly the internal 7x7 square (color cells) (Square A4). For example in row 3, 21 is added to 77 and 26 to 72. In column 9,
27 is added to 71 and 28 to 70, followed by their complements, using numeric superscripts.
 Fill in similarly the internal 9x9 square (color cells) (Square A5). For example in row 2, 23 is added to 84, 24 to 83 and 22 to 86 using numeric
superscripts. In column 10, 31 is added to 76, 32 to 75 and 41 to 67 followed by their complements, using numeric superscripts.
 Finally fill in the external 11x11 square (color cells) (Square A6). For example in row 1, 25 is added to 78,
29 to 74, 34 to 87 and 42 to 79. In column 11, 30 is added to 73, 40 to 63, 53 to 68 and 57 to 64 along with their complements using numeric superscripts.
 No connectivity between numbers using complicated spaghetti lines will be attempted but will now be replaced by the coded table and by the color complementary
table shown at the end.
 Below is the coded connections to this square where the colored "spoke"cells are
not included in the coding:
4  ...  21  22  23  24  25  26  27  28  29  30 
31  32  33  34  35  36  37  38  39 

118  ...  101  100  99  98  97  96  95  94  93  92 
91  90  89  88  87  86  85  84  83 


53^{1}  ...  25^{1}  15^{1}  16^{1}  16^{2}  20^{1} 
25^{2}  25^{3}  25^{4} 
20^{2}  20^{3}  16^{3}  16^{4}  5^{a} 
2^{1}  2^{1}  15^{1}  5^{a}  16^{1}  16^{2} 


40  41  42  43  44 
45  46  47  48  49  50  51  52 
53  54  55  56  57  58  59  60 
 61 
82  81  80  79  78 
77  76  75  74  73  72  71  70 
69  68  67  66  65  64  63  62 


20^{4}  15^{2}  2^{2}  2^{2}  20^{1} 
25^{1}  16^{3}  16^{4}  20^{2}  20^{3} 
25^{2}  25^{3}  25^{4} 
2^{3}  2^{3}  15^{2}  53^{1} 
2^{4}  2^{4}  20^{4} 
 The coded method as well as the color squares (below Square A7) will be used from here on due to excessive crisscrossing of lines.
 Square A6 shows the 5 border squares in "border format".
 The complement table below also shows how the color pairs are layed out (for comparison with Square A6).
A1
104     
17    
 102 
 108    
13     106 

  112   
9    110 
 
   116  
5   114  
 
    120 
1  62 
   
19  15  11 
7  3 
61  119 
115  111 
107  103 
    60 
121  2 
   
   8  
117   6  
 
  12   
113    10 
 
 16    
109     14 

20     
105    
 18 

⇒ 
A2 (Δ=1,δ=4)
104     
17    
 102  448 
2(103+121) 
 108    
13     106 
 322  107x2+108 
  112   
9    110 
  196  98x2 
   116  
5   114  
  70  70 
    120 
1  62 
     
19  15  11 
7  3 
61  119 
115  111 
107  103 
 
    60 
121  2 
    

   8  
117   6  
  174 
174 
  12   
113    10 
  292 
2x146 
 16    
109     14 
 402  137x2+136 
20     
105    
 18  528 
2(123+141) 
528  410  292 
174   
 70 
196  322  448 
 

⇒ 
A3
104     
17    
 102 
 108    
13     106 

  112   
9    110 


   116  4 
5  66 
114    
   89  120 
1  62 
33    
19  15  11 
7  3 
61  119 
115  111 
107  103 
   85  60 
121  2 
37    
   8  118 
117  56 
6    
  12   
113   
10   
 16    
109     14 

20     
105     
18 

 ⇒ 
A4
104     
17    
 102 
 108    
13     106 

  112  21 
26  9  72 
77  110   
  95 
116  4 
5  66 
114  27   
  94  89  120 
1  62 
33  28   
19  15  11 
7  3 
61  119 
115  111 
107  103 
  52  85  60 
121  2  37 
70   
  51  8 
118 
117  56 
6  71   
  12  101 
96  113 
50  45 
10   
 16    
109     14 

20     
105     
18 

⇒ 
A5
104     
17    
 102 
 108  23 
24  22 
13  86 
83  84 
106  
 91  112 
21  26 
9  72 
77  110 
31  
 90  95 
116  4 
5  66 
114  27 
32  
 81  94 
89  120 
1  62 
33  28  41  
19  15  11 
7  3 
61  119 
115  111 
107  103 
 55  52  85 
60  121 
2  37 
70  67  
 47  51 
8  118 
117  56 
6  71 
75  
 46 
12  101 
96  113 
50  45 
10  76  
 16  99 
98  100 
109  36  39 
38  14  
20     
105     
18 

⇒ 
A6
104  25 
29  34 
42  17 
79  87 
74  78 
102 
92  108  23 
24  22 
13  86 
83  84 
106  30 
82  91  112 
21  26 
9  72 
77  110 
31  40 
69  90  95 
116  4 
5  66 
114  27 
32  53 
65  81  94 
89  120 
1  62 
33  28  41 
57 
19  15  11 
7  3 
61  119 
115  111 
107  103 
58  55  52  85 
60  121 
2  37 
70  67  64 
54  47  51 
8  118 
117  56 
6  71 
75  68 
59  46 
12  101 
96  113 
50  45 
10  76 
63 
49  16  99 
98  100 
109  36  39 
38  14  73 
20  97 
93  88 
80  105 
43  35 
48  44 
18 

⇒ 
A7
104  25 
29 
34  42 
17  79 
87  74 
78  102 
92  108 
23 
24  22 
13  86 
83  84 
106  30 
82  91 
112 
21  26 
9  72 
77  110 
31  40 
69  90 
95 
116  4 
5  66 
114  27 
32  53 
65  81 
94  89 
120  1  62 
33  28  41 
57 
19  15 
11  7 
3  61  119 
115  111  107 
103 
58  55 
52  85 
60  121  2 
37  70  67 
64 
54  47 
51  8 
118  117 
56  6 
71  75 
68 
59  46 
12  101 
96  113 
50  45 
10  76 
63 
49  16 
99 
98  100 
109  36 
39  38 
14  73 
20  97 
93  88 
80  105 
43  35 
48  44 
18 
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 
41 
 
101 
100  99 
98  97 
96  95 
94  93 
92  91 
90  89 
88  87 
86  85  84 
83  82 
81 
42  43 
44  45  46 
47  48  49 
50  51 
52  53  54 
55  56  57 
58  59  60 
 61 
80  79 
78  77  76 
75  74  73 
72  71 
70  69  68 
67  66  65 
64  63  62 
This completes the Part W5 of a 11x11 Magic Square Wheel Spoke Shift method. To go to Part W6 of an 11x11 square.
Go back to homepage.
Copyright © 2015 by Eddie N Gutierrez