A New Procedure for Magic Squares (Part IB)
Centered Sequential MaskGenerated Squares
A Discussion of the New Method
Skip the discussion go to examples
In this method the numbers on a 4n + 1 square are placed consecutively starting from the center cell in the top row and entered
until every other cell is filled.
Consecutive numbers are then added to the next rows using a (1,down,1,right) knight move as shown below in the examples. However, there are two exceptions to the rule:
(1) the center row is not filled in until until all the other rows are partially filled in and
(2) the row adjacent to the center row (on going down the square) has its entries filled in a different
order, i.e., not using the (1,down,1,right) knight move.
Additions to the square going up are performed, first filling in the center row then filling in the empty cells.
Note that since two types of odd squares exists, i.e., the 4n + 1 and the 4n + 3, two methods for filling in the
squares are possible. The former example using
n = 5 and 9 are shown below.
After converting the squares into semimagic ones the square are converted into magic ones by the use of a mask. This mask generates numbers
which are added to certain cells in the square to produce a final square composed of numbers which may not be in serial order. For example, negative numbers or numbers greater
than n^{2} may be present in the square.
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 Readout  use of a mask
 Construct the 9x9 Square 1 where 9 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 3 and filling every other
the cell (square 6).
 Upon reaching the numeral 5 perform a knight break enter 6 at cell 4 and continue filling cells in a normal fashion.
 After performing exception (2) skip over center row and add 19 to the second cell in the 6th row and continue adding numerals in normal fashion in the last row of the
square.
 From 36 go to the center row and fill in the row consecutively (square 7).
6
4   5   1 
 2   3 
 9   6  
7   8  
14   10   11 
 12   13 
 18   17   16 
 15  
    
  
 19   20  
21   22  
24   25   26 
 27   23 
 29   30  
31   28  
34   35   36 
 32   33 

⇒ 
7
4   5   1 
 2   3 
 9   6  
7   8  
14   10   11 
 12   13 
 18   17   16 
 15  
37  38  39  40  41 
42  43  44  45 
 19   20  
21   22  
24   25   26 
 27   23 
 29   30  
31   28  
34   35   36 
 32   33 

⇒ 
 Fill in the rest of the square, going backwards this time (square 8), starting at
45 → 46. At 63 go to 64 adjacent to the numeral 2 on the first row.
 Since only the columns are equal to the magic sum while the row and diagonal sums are not, add or subtract the numbers in the last column to those of the center column.
At this point four duplicates have been generated in pink (Square 9).
8
 188  
4  66  5  67  1 
64  2  65  3  277  92 
71  9  72  6  68 
7  69  8  70  380  11 
14  76  10  73  11 
74  12  75  13  358  11 
81  18  77  17  78  16 
79  15  80  461  92 
37  38  39  40  41 
42  43  44  45  369  0 
47  19  48  20  49 
21  50  22  46  322  47 
24  52  25  53  26 
54  27  51  23  335  34 
57  29  58  30  59 
31  55  28  56  403  34 
34  62  35  63  36 
60  32  61  33  416  47 
369  369  369 
369  369  369 
369  369  369 
190  

⇒ 
9
 188 
4  66  5  67  93 
64  2  65  3  369 
71  9  72  6  57 
7  69  8  70  369 
14  76  10  73  22 
74  12  75  13  369 
81  18  77  17  14  16 
79  15  80  369 
37  38  39  40  41 
42  43  44  45  369 
47  19  48  20  96 
21  50  22  46  369 
24  52  25  53  60 
54  27  51  23  369 
57  29  58  30  25 
31  55  28  56  369 
34  62  35  63  11 
60  32  61  33  369 
369  369  369 
369  369  369 
369  369  369 
190 

+ 
 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 3 (as in the de la Hire method) that all sums will equal.
 We start by subtracting the diagonals(188,190) from 369 to give 181 and 179, respectively and which will be used as what I call the
"de la Hire constants".
Addition of the sum of these two numbers, 181 + 179 = 360 to
369 gives 729 a magic presum. This sum is just right for our purpose.
 The following equations are used such that
the following conditions are obeyed:
The right diagonal: 729 = 188 + 2(181) + 179
The left diagonal: 729 = 190 + 181 + 2(179)
The rows and columns: 729 = 369 + 181 + 179
 Generate the mask using the 181 and 179 factors and adding these factors to the appropriate cells in square 9 to generate square 10.
 Square 10 has a magic sum equal to 729, i.e., S = 729 = ½(n^{3} + 81n).
Mask B
  
181      179 
 179    181  
  
181   179  
    
   
179  181    
  
   360   
  
    179  181 
  181 
179      
179   
    181  
 181  
  179    

⇒ 
10
 729 
4  66  5  248  93  64 
2  65  182  729 
71  188  72 
6  238  7  69  8  70  729 
195  76  189  73  22 
74  12  75  13  729 
81  18  77  17  165 
197  79  15  80  729 
37  38  39  40  41 
42  403  44  45  729 
47  19  48  20  96 
21  50  201  227  729 
24  52  206  232  60 
54  27  51  23  729 
236  29  58  30  25 
31  55  209  56  729 
34  243  35  63  11 
239  32  61  33  729 
729  729  729 
729  729  729 
729  729 
729  729 

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