A New Procedure for Magic Squares (Part IA)
Equation Generated 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. (This is a modification of the 5x5 square constructed previously ).
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 is an exception to the rule:
(1) the center row is not filled in until until all the other rows are partially filled in.
Additions to the square going up are performed by filling in the center row followed by the empty cells below the center row. The cells above the filled centered row
are filled according to the general sequence formula shown below. The only exception being that when n = 5 only one move is possible (3),
i.e., 3 cells to the left as shown in the sequence table.
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)
Table S is generated by using the following general sequence:
{ (2), [½(n  1), (2)_{r},(3)] }
where n is an odd number of the type 4m + 1 and r is a repeating sequence equal to
½(n  9).


Sequence Table
Number of cells to move per break  Direction of row fillings 
(3)  R 
(2, 4, 3)  R, R, R 
(2, 6, 2, 2, 3)  R, R, R, R, R 
(2, 8, 2, 2, 2, 2, 3)  R, R, R, R, R, R, R 
(2, 10, 2, 2, 2, 2, 2, 2, 3)  R, R, R, R, R, R, R, R, R 

Method: Sequential Readout  use of a mask
 Construct the 5x5 Square 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 knight break (1 down, 1 right) and enter a 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 
 4  
5  
11  12  13 
14  15 
 7  
6  
10   8 
 9 

⇒ 
3
 37  
3  22  1 
21  2  49  16 
24  4  25 
5  23  81  16 
11  12  13 
14  15  65  0 
17  7  18 
6  16  64  1 
10  20  8 
19  9  66  1 
65  65  65 
65  65  35  

⇒ 
4
 37 
3  22  17 
21  2  65 
24  4  9 
5  23  65 
11  12  13 
14  15  65 
17  7  19 
6  16  65 
10  20  7 
19  9  65 
65  65  65 
65  65  35 

 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(35,37) from 65 to give 30 and 28, respectively and which will be used as what I call the
"de la Hire constants".
Addition of 30 and 28 to 65 gives 123 a magic presum. However,
 The following equations are used such that the following conditions are obeyed:
The right diagonal: 123 = 37 + 2(28) + 30
The left diagonal: 123 = 35 + 28 + 2(30)
The rows and columns: 123 = 65 + 28 + 30
 Generate the mask using the 28 and 30 factors or sums thereof (58) adding these factors to the appropriate cells in square 4 to generate square 5.
 Square 5 has a magic sum equal to 123, i.e., S = 123 = ½(n^{3} + 24n +
1).
4
 37 
3  22  17 
21  2  65 
24  4  9 
5  23  65 
11  12  13 
14  15  65 
17  7  19 
6  16  65 
10  20  7 
19  9  65 
65  65  65 
65  65  35 

+ 
Mask A
30   28 
 
 
 58  
28  30  
 
 28  30 
 
  
 58 

⇒ 
5
 123 
33  22  45 
21  2  123 
24  4  9 
63  23  123 
39  42  13 
14  15  123 
17  35  49 
6  16  123 
10  20  7 
19  67  123 
123  123  123 
123  123  123 

This completes this section on a new Equation Generated Centered Sequential MaskGenerated Squares (Part IA). The next section deals with
Equation Generated Centered Sequential 9x9 MaskGenerated Squares (Part IB). To return to homepage.
Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com