NEW MAGIC SQUARES WHEEL METHOD  BORDER SQUARES
Part X4
How to generate 11x11 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 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).
This site introduces a new method used for the construction of border wheel type squares. The method consists of forming a 7x7 internal Wheel
magic square then filling in the external 1,2 and 10,11 rows and columns with the requisite nonspoke 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 
 
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, the symbol δ (where δ = 4) specifies the difference between entries on the diagonals and center row and column not situated on the
center 7x7 square (shown in square A1)
The nonspoke entries in rows and columns 1,2 and 10,11 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 11x11 Transposed Magic Square Using the Diagonals {32,36,58,59,60,61,62,63,64,86,90} and {30,34,56,39,78,61,44,83,66,88,92}
 Fill in the internal 7x7 square with the numbers 37 through 85 according to the wheel method to form a magic square.
 Subtract δ = 4 from 37 and add this number (33) to the center cell of row 2 and repeat again (334) = 29 and place this number in the center cell of row 1.
Starting with the number 29 place consecutive numbers as shown in Square A1 in a spiraling fashion up to the number 36, followed by their complementary
numbers (Square A1). This completes the spokes for the square, along with the nonspoke numbers of the 7x7 square.
 Thus the spokes of the wheel are shown as follows: Left diagonal 32,36...86,90; right diagonal 30,34...88,92; central column 29,33...89,93; central row
31,35...87,91. (...) denotes the numbers from the 7x7 square (Square A1).
 Sum up the rows with the diagonal and central row or column and subtract from 549 (sum of 9x9 internal square), to give the amounts required to complete
the 9x9 square. The 13^{th} rows shows the numbers required. (Square A2).
 Fill in the required pairs for the row and columns chosen from the coded table below coded in numeric superscripts.
 Repeat for rows and columns 1,11. However, subtract the numbers from 671 (the sum of a 11x11 magic square). Square A2 shows 4 pairs of numbers are required
and these are listed down. The coded table below shows which numbers pair up.
Note that the sum of the numbers in rows and columns 1 and 11 sum up to 122 (Square A4).
 Below is the coded connections to this square where the colored "spoke" cells and those labeled (...)
i.e., the 7x7 entries, are not included in the coding.
1  2  3  4  5  6  7  8  9  10  11  12 
13  14  15  16 

121  120  119  118  117  116  115  114  113  112  111  110 
109  108  107  106 


26^{1}  9^{1}  26^{2}  9^{2}  9^{3}  9^{4} 
9^{5}  9^{6}  3^{1}  9^{1}  3^{1}  9^{2}  9^{3} 
9^{4}  9^{5}  9^{6} 


17  18  19  20  21  22  23  24  25 
26  27  28  ...  58 
59  60 
 61 
105  104  103  102  101  100  99  98  97 
96  95  94  ...  64 
63  62 


3^{2}  3^{3}  3^{2}  3^{3} 
2^{1}  2^{1}  2^{2}  2^{2} 
3^{4}  26^{1}  3^{4}  26^{2} 


 Square A5 shows the 3 border squares in "border format".
 The complementary table below also shows how the color pairs of Square A4 are layed out.
A1
92     
29    
 90 
 88    
33     86 

  82  50 
53  37  71 
70  64   
  54  81 
74  38  49 
63  68   
  57  48 
80  39  62 
76  65   
31  35  43 
44  45 
61  77 
78  79 
87  91 
  67  73 
60  83  42 
47  55   
  66  59 
46  84  75 
41  56   
  58  72 
69  85  51 
52  40   
 36    
89     34 

32     
93    
 30 

⇒ 
A2
92     
29    
 90  460 
123+97+(120x2) 
 88    
33     86 
 342  114x3 
  82  50 
53  37  71 
70  64   
 
  54  81 
74  38  49 
63  68   
 
  57  48 
80  39  62 
76  65   
 
31  35  43 
44  45 
61  77 
78  79 
87  91 
 
  67  73 
60  83  42 
47  55   
 
  66  59 
46  84  75 
41  56   
 
  58  72 
69  85  51 
52  40   
 
 36    
89     34 
 390  130x3 
32     
93    
 30  516 
121+(121x2)+147 
516  390  
  
 
 342  460 
 

⇒ 
A3
92     
29    
 90 
 88  2 
4  5 
33  109 
110  112 
86  
 116  82 
50  53 
37  71 
70  64 
6  
 115  54 
81  74 
38  49 
63  68 
7  
 114  57 
48  80 
39  62 
76  65  8  
31  35  43 
44  45 
61  77 
78  79 
87  91 
 16  67  73 
60  83 
42  47 
55  106  
 15  66 
59  46 
84  75 
41  56 
107  
 14 
58  72 
69  85 
51  52 
40  108  
 36  120 
118  117 
89  13  12 
10  34  
32     
93     
30 

⇒ 
A4
92  22 
9  17 
1  29 
96  103 
111  101 
90 
98  88  2 
4  5 
33  109 
110  112 
86  24 
104  116  82 
50  53 
37  71 
70  64 
6  18 
97  115  54 
81  74 
38  49 
63  68 
7  25 
119  114  57 
48  80 
39  62 
76  65  8 
3 
31  35  43 
44  45 
61  77 
78  79 
87  91 
28  16  67 
73  60  83 
42  47 
55  106  94 
27  15  66 
59  46 
84  75 
41  56 
107  95 
20  14 
58  72 
69  85 
51  52 
40  108 
102 
23  36 
120  118 
117  89 
13  12 
10  34 
99 
32  100 
113  105 
121  93 
26  19 
11  21 
30 

⇒ 
A5 Partial Border
92  22 
9 
17  1 
29  96 
103  111 
101  90 
98  88 
2 
4  5 
33  109 
110  112 
86  24 
104  116 
82 
50  53 
37  71 
70  64 
6  18 
97  115 
54 
81  74 
38  49 
63  68 
7  25 
119  114 
57  48  80 
39  62 
76  65 
8  3 
31  35 
43  44 
45  61  77 
78  79  87 
91 
28  16 
67  73 
60  83  42 
47  55  106 
94 
27  15 
66  59 
46  84 
75  41 
56  107 
95 
20  14 
58  72 
69  85 
51  52 
40  108 
102 
23  36 
120 
118  117 
89  13 
12  10 
34  99 
32  100 
113  105 
121  93 
26  19 
11  21 
30 
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 X4 of a 11x11 Magic Square Wheel Spoke Shift method.
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Copyright © 2015 by Eddie N Gutierrez