A New Procedure for Magic Squares (Part IA)
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 5x5 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 3 and filling every other
the cell (square 1).
 Upon reaching the numeral 3 (this row is adjacent to the center row so no knight break is done) enter 4 at cell 4 and continue filling cells in a normal fashion.
 Skip over center row and add 6 two cells down from the 5 and continue adding numerals in normal fashion breaking at 7.
 From 10 go to the center row and fill in the row consecutively (square 2). Fill in the rest of the square, at 20 adding the next numeral to the right of 1 (square 3).
 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 (Square 4).

⇒ 
2
3   1 
 2 
 5  
4  
11  12  13 
14  15 
 6  
7  
9   10 
 8 

⇒ 
3
 34  
3  22  1 
21  2  49  16 
25  5  23 
4  24  81  16 
11  12  13 
14  15  65  0 
17  6  18 
7  16  64  1 
9  20  10 
19  8  66  1 
65  65  65 
65  65  36  

⇒ 
4
 34 
3  22  17 
21  2  65 
25  5  7 
4  24  65 
11  12  13 
14  15  65 
17  6  19 
7  16  65 
9  20  9 
19  8  65 
65  65  65 
65  65  36 

 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 the diagonals(34,36) from 65 to give 31 and 29, respectively and which will be used as what I call the
"de la Hire constants".
Addition of 31 and 29 to 65 gives 125 a magic presum. However, 154 = 125 + 29 turns out to be a better choice.
 The following equations are used such that the following conditions are obeyed:
The right diagonal: 154 = 34 + 2(29) +2(31)
The left diagonal: 154 = 36 + 3(29) + 31
The rows and columns: 154 = 65 + 29 + 2(31).
 Generate the mask using the 29 and 31 factors or sums thereof (60) adding these factors to the appropriate cells in square 4 to generate square 5.
 Square 5 has a magic sum equal to 125, i.e., S = 154 = ½(n^{3} + 36n +
3).
4
 34 
3  22  17 
21  2  65 
25  5  7 
4  24  65 
11  12  13 
14  15  65 
17  6  19 
7  16  65 
9  20  9 
19  8  65 
65  65  65 
65  65  36 

+ 
Mask A
 29  60 
 
29  29 
  31 
 31  29 
 29 
  
60  29 
60   
29  

⇒ 
5
 154 
3  51  77 
21  2  154 
54  34  7 
4  55  154 
11  43  42 
14  44  154 
17  6  19 
67  45  154 
69  20  9 
48  8  154 
154  154  154 
154  154  154 

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