A New Procedure for Magic Squares (Part IE)
Centered Sequential Zig Zag 11x11 MaskGenerated Squares
A Discussion of the New Method
Skip the discussion go to examples
In this method the numbers on a 4n + 3 square are placed consecutively starting from the center cell in the top row and entered
in a zig zag manner. Additions to the square are done by leaving the center row unfilled and continuing the zig zag pattern below the center row. (The addition across the center row is different
from previous additions which began on the first cell. The addition of numbers in this case may come elsewhere)). The center row is then
filled in consecutively followed by the filling in of the bottom and top rows again in a zig zag manner. Breaking involves moving down 2 cells to the next row and
continuing addition of numbers to the right.
After the square is filled, the square is converted into a semimagic one by addition or subtraction i,e. by the differences of numbers in the
last row. The square is then converted by means of a numerical mask into a magic square. Moreover, this new square may have negative numbers or numbers greater than
n^{2}.
In addition, it will also be shown that the sums of these squares follows either of the two sum equations shown in the
New block Loubère Method and Consecutive 5x5 Mask Generated squares:
S = ½(n^{3} ± an)
S = ½(n^{3} ± an + b)
Method: Sequential Zig Zag Readout  use of a mask
 Construct the 11x11 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 5 and filling every other cell (square 1). Note that
on reaching the final cell in the row the addition is continued at the other end of the row.
 Upon reaching 11 move down two cells to 12 and continue in the same manner eventually skipping over the center row.
 From 55 go to the center row and fill in the row consecutively (square 2).
1
 8   10  
1   3   5  
7   9   11  
2   4   6 
 20   22  
13   15   17  
19   21   12  
14   16   18 
 32   23  
25   27   29  
    
     
31   33   24 
 26   28   30 
 44   35  
37   39   41  
43   34   36 
 38   40   42 
 45   47  
49   51   53  
55   46   48 
 50   52   54 

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

⇒ 
 From 66 go down two cells to 67 and repeat the zig zag pattern (Square 3).
 From 88 move up to 89 and add numbers zigzag across the center row to the right.
 At 99 go a distance of five units up to 100 and continue adding in zigzag fashion to fill in square 4.
 At this point all columns at this point sum to 671,
while the row sums are to be adjusted (by + or  values) according to the last cell in square 4 in order to sum to 671.
3
 8   10  
1   3   5  
7   9   11  
2   4   6 
 20   22  
13   15   17  
19   21   12  
14   16   18 
92  32  94  23  96 
25  98  27  89  29  91 
56  57  58  59  60 
61  62  63  64  65  66 
31  93  33  95  24 
97  26  99  28  90  30 
68  44  70  35  72 
37  74  39  76  41  67 
43  69  34  71  36 
73  38  75  40  77  42 
80  45  82  47  84 
49  86  51  88  53  79 
55  81  46  83  48 
85  50  87  52  78  54 

⇒ 
4
 782  
104  8  106  10  108 
1  110  3  101  5  103  659  23 
7  105  9  107  11  109 
2  100  4  102  6  562  109 
116  20  118  22  120 
13  111  15  113  17  115  880  109 
19  117  21  119  12  121 
14  112  16  114  18  683  12 
92  32  94  23  96  25  98  27 
89  29  91  696  25 
56  57  58  59  60 
61  62  63  64  65  66  671  0 
31  93  33  95  24 
97  26  99  28  90  30  646  25 
68  44  70  35  72 
37  74  39  76  41  67  623  48 
43  69  34  71  36 
73  38  75  40  77  42  598  73 
80  45  82  47  84  49  86 
51  88  53  79  744  73 
55  81  46  83  48 
85  50  87  52  78  54  719  48 
671  671  671 
671  671  671 
671  671  671 
671  671  815  

⇒ 
 Square 5 shows the result of the adjustment with the generation of 0 duplicates.
5
 782 
104  8  106  10  108 
13  110  3  101  5  103  671 
7  105  9  107  11  218 
2  100  4  102  6  671 
116  20  118  22  120 
96  111  15  113  17  115  671 
19  117  21  119  12  109 
14  112  16  114  18  671 
92  32  94  23  96  0  98  27 
89  29  91  671 
56  57  58  59  60  61 
62  63  64  65  66  671 
31  93  33  95  24 
122  26  99  28  90  30  671 
68  44  70  35  72 
85  74  39  76  41  67  671 
43  69  34  71  36 
146  38  75  40  77  42  671 
80  45  82  47  84  24  86 
51  88  53  79  671 
55  81  46  83  48 
37  50  87  52  78  54  671 
671  671  671 
671  671  671 
671  671  671 
671  671  815 
 Generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each
of these values is added to the corresponding cell in square 4 (as in the de la Hire method) that all sums will be equal.
 We start by subtracting 671 from the diagonals(782,815) to give 111 and 144, respectively, and which will be used as what I call the
"de la Hire constants".
Addition of 111 and 144 to 671 gives 926 a magic presum.
 The following equations are used such that the following conditions are obeyed:
The right diagonal: 926 = 782 + 144
The left diagonal: 926 = 815 + 111
The rows and columns: 282 = 175 + 50 + 57.
 Generate the mask using the 111 and 144 factors or sums thereof adding these factors to the appropriate cells in square 5 to generate square 6.
 Square 6 has a magic sum equal to 926, i.e., S = 926 = ½(n^{3} + 47n +
4).
Square 5  + 
Mask A
  111     144 
   
 111      
 144   
   111    
   144 
     111  
144    
     144  
  111  
 144    111   
   
      
  144  111 
111   144     
   
144       
111    
   144    111 
   
    144   
 111   

⇒ 
6
 926 
104  8  217  10  108 
13  254  3  101  5  103  926 
7  216  9  107  11  218 
2  100  148  102  6  926 
116  20  118  133  120 
96  111  15  113  17  259  926 
19  117  21  119  12  220 
14  256  16  114  18  926 
92  32  94  23  96  144  98  27 
89  140  91  926 
56  201  58  59  171  61 
62  63  64  65  66  926 
31  93  33  95  24  122  26 
99  28  234  141  926 
179  44  214  35  72 
85  74  39  76  41  67  926 
187  69  34  71  36 
146  38  186  40  77  42  926 
80  45  82  191  84  24  197 
51  88  53  79  926 
55  81  46  83  192 
37  50  87  163  78  54  926 
926  926  926 
926  926  926 
926  926  926 
926  926  926 

This completes this section on a new Centered Sequential MaskGenerated Squares (Part IC). To return to homepage.
Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com