A New Procedure for Magic Squares (Part I)
Consecutive Internally Added MaskGenerated Squares
A Discussion of the New Method
Magic squares such as the Loubère have a center cell which must always contain the middle number of
a series of consecutive numbers, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or
½(n^{2} + 1). The properties of these regular or associated Loubère squares are:
 That the sum of the horizontal rows,
vertical columns and corner diagonals are equal to the magic sum S.
 The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to
n^{2} + 1, i.e., or twice the number in the center cell and are complementary to each other.
In this method the numbers on the square are placed consecutively starting from the second leftmost column and entered across every other cell. Consecutive numbers
are then added to the next rows boustrophedonically. When n > 7 at least one row turns into a regular left to right order, increasing by one
for each n. In addition every other number in the center row (starting with the second cell) will take on its complement. For example for a 5x5
square the second number 12 becomes 14 and 14 becomes 12 (see the 5x5 example below).
The final square is 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 follow the sum equation shown in the
New block Loubère Method. :
S = ½(n^{3} ± an)
Construction of a 5x5 Magic Square
Method: Reading boustrophedonically (like a sidewinder snake)  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. Don't fill in the center row
but proceed to the first cell in the fourth row (the number 6).
 On reaching 10 reverse the pattern by adding consecutive numbers (remembering that every other number except for the center cell takes
on its complement) filling the center row then proceeding from 15 to 16 along the
yellow path, and filling in the last two rows. On reaching 20 proceed to 21 and fill up the top two rows consecutively
(Square 4).

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

⇒ 
3
 1  
2  
5   4 
 3 
11  14  13 
12  15 
6  16  7 
17  8 
20  10  19 
9  18 

⇒ 
4
 95  
23  1  22 
2  21  69  4 
5  24  4 
25  3  61  4 
11  14  13 
12  15  54  0 
6  16  7 
17  8  76  11 
20  10  19 
9  18  66  11 
65  65  65 
65  65  95  

⇒ 
 Since the columns are all equal to 65 add or subtract the numbers in the last row from the center column values to generate square 5.
At this point two duplicates have been generated.
 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 equal 120.
 We start by subtracting 65 from each of the diagonals(95,95) (square 5) to give 30 and 30, respectively and which will be used as what I call the
"de la Hire constants".
Addition of 30 to 65 gives 95 a magic presum. Since two numbers from the center column must be changed another number
must be added to 65. The number we use is n^{2} = 25.
 The following equations are used such that the following conditions are obeyed:
The right diagonal: 120 = 95 + 25
The left diagonal: 120 = 95 + 25
The rows and columns: 120 = 65 + 25 + 30.
 Generate the mask using the 25 and 30 factors adding these factors to the appropriate cells in square 5 to generate square 6.
 Square 6 has a magic sum equal to 120, i.e., S = 120 = ½(n^{3} + 23n).
5
 95 
23  1  18 
2  21  65 
5  24  8 
25  3  65 
11  14  13 
12  15  65 
6  16  18 
17  8  65 
20  10  8 
9  18  65 
65  65  65 
65  65  95 

+ 
Mask A
  30 
25  
30   25 
 
25   
30  
 25  
 30 
 30  
 25 

⇒ 
6
 120 
23  1  48 
27  21  120 
35  24  33 
25  3  120 
36  14  13 
42  15  120 
6  41  18 
17  38  120 
20  40  8 
9  43  120 
120  120  120 
120  120  120 

Construction of a 7x7 Magic Square
Method: Reading consecutive from left to right boustrophedonically  use of a mask
 Construct the 7x7 Square 1 where 7 = 4n + 3 by adding consecutive numbers in a consecutive manner to the cells.At the number 8 proceed
in a zig zag manner to number 14 then to the 5th row second cell and enter 9. The center row is not filled at this time.
 On reaching 21 reverse the pattern by adding consecutive numbers 2228 to the center row (remembering that every other number except for the center
cell takes on its complement)
yellow and khaki path, then filling in the last two rows.
On reaching 35 proceed to 36 and fill in the middle rows reversibly in a zig zag manner.
7
 1  
2   3  
7   6 
 5   4 
8   10 
 12   14 
  
   
 9  
11   13  
18   17 
 16   15 
 19  
20   21  

⇒ 
8
 1  
2   3  
7   6 
 5   4 
8   10 
 12   14 
22  27  24 
25  26  23  28 
 9  
11   13  
18  31  17 
30  16  29  15 
32  19  33 
20  34  21  35 

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

⇒ 
 Fill in the last top rows as shown in square 10.
 Since the columns are all equal to 175 add or subtract the numbers in the last row from the center column values to generate square 11.
At this point two duplicates have been generated (Square 11).
10
 232  
46  1  45  2  44  3 
43  184  9 
7  47  6 
48  5  49  4  166  9 
8  41  10 
39  12  37  14  161  14 
22  27  24 
25  26  23  28  175  0 
42  9  40 
11  38  13  36  189  14 
18  31  17 
30  16  29  15  156  19 
32  19  33 
20  34  21  35  194  19 
175  175  175 
175  175  175 
175  230  

⇒ 
11
 232 
46  1  45  7  44  3 
43  175 
7  47  6 
57  5  49  4  175 
8  41  10 
53  12  37  14  175 
22  27  24 
25  26  23  28  175 
42  9  40 
3  38  13  36  175 
18  31  17 
49  16  29  15  175 
32  19  33 
1  34  21  35  175 
175  175  175 
175  175  175 
175  230 

+ 
 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 10 (as in the de la Hire method) that all sums will equal 287.
 We start by subtracting 175 from each of the diagonals(230,232) (square 9) to give 55 and 57, respectively and which will be used as what I call the
"de la Hire constants".
Addition of the sum of these two numbers, 55 + 57 = 112 to
175 gives 287 a magic presum.
 The following equations are used such that
the following conditions are obeyed:
The right diagonal: 287 = 230 + 57
The left diagonal: 287 = 232 + 55
The rows and columns: 287 = 175 + 55 + 57
 Generate the mask using the 55 and 57 factors and adding these factors to the appropriate cells in square 10 to generate square 11.
 Square 5 has a magic sum equal to 287, i.e., S = 287 = ½(n^{3} + 33n).
Mask B
  57 
 55   
 57  55 
   
57   
   55 
55   
 57   
  
  55  57 
 55  
57    
  
55   57  

⇒ 
12
 287 
46  1  102  7  99  3 
43  287 
7  104  61 
57  5  49  4  287 
65  41  10 
53  12  37  69  287 
77  27  24 
25  83  23  28  287 
42  9  40 
3  38  68  93  287 
18  86  17 
106  16  29  15  287 
32  19  33 
56  34  78  35  287 
287  287  287 
287  287  287 
287  287 

Note that if square 9 is produced by swithching several entries in the last two rows that a cross square is generated. All numbers in the center row are 25 and
the average of the sums of the 1^{st} and 2^{nd}, 3^{rd} and 5^{th} and 6^{th} and 7^{th} are each 25.
13
 232  
46  1  45  2  44  3 
43  184  9 
7  47  6 
48  5  49  4  166  9 
8  41  10 
39  12  37  14  161  14 
22  27  24 
25  26  23  28  175  0 
42  9  40 
11  38  13  36  189  14 
15  31  16 
30  17  29  18  156  19 
32  21  33 
20  34  19  35  194  19 
172  177  174 
175  176  173 
178  230  
3  2  1 
0  1  2  3   

⇒ 
14
 232 
46  1  45  7  44  3 
43  184 
7  47  6 
57  5  49  4  166 
8  41  10 
53  12  37  14  161 
25  25  25 
25  25  25 
25  175 
42  9  40 
3  38  13  36  189 
15  31  16 
49  17  29  18  156 
32  21  33 
1  34  19  35  194 
175  175  175 
175  175  175 
175  230  

This completes this section on a new Consecutive Internally Added MaskGenerated Squares (Part I). The next section deals with
Consecutive Internal 9x9 MaskGenerated Squares (Part II). To return to homepage.
Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com