A New Procedure for Magic Squares
A Loubère Type Method  The Slant Break
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 added by the staircase method starting at (one down, one right) to the center cell. The Loubère method is applied
(the staircase method). When a break is encountered the move is (2 down, 2 right), i.e., a 2 down slant. The result is the generation of a Loubère type
square different from the standard Loubère.
In addition, those numbers divisible by three will not be magic but will give triads of sums, viz. for a 9 x9 this triad is (360,369,379).
To convert this square to magic we can use the mask method as shown in Cross and Mask Methods.
The sums of these squares is also shown to follow the new sum equation as was shown in the
New block Loubère Method:
S = ½(n^{3} ± an)
Construction of a 5x5 Slant Break Magic Square
 Construct Square 1 by adding consecutive numbers at position (one down, one right) to the center cell.
On reaching 5 break (2 down,2 right) and continue with 6.
 Fill in the rest of the cell in like manner.

⇒ 
2
7  4  24 
18  15 
3  25  17 
14  6 
24  16  13 
10  2 
20  12  9 
1  23 
11  8  5 
22  19 

Construction of a slant break 9x9 Magic Square
Use of a mask
 Construct Square 1 by adding consecutive numbers at position (one down, one right) to the center cell.
On reaching 5 break (2 down,2 right) and continue with 10.
 Fill in the rest of the cell in like manner (Squares 2 and 3).
 At this point not all rows sum to 369.
1
 6     18 
  
5     17 
   
   16  
  19  4 
  15   
  3  
 14    
 2   
13     
1    
    9 
   12 
   8  
  11  
  7   
 10   

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

⇒ 
3
 369  
21  6  72  48  33  18 
75  60  45  378  9 
5  71  47  32  17 
74  59  44  20  369  0 
70  46  31  16  73 
58  43  19  4  360  9 
54  30  15  81  57 
42  27  3  69  378  9 
29  14  80  56  41 
26  2  68  53  369  0 
13  79  55  40  25 
1  67  52  28  360  9 
78  63  39  24  9 
66  51  36  12  378  9 
62  38  23  8  65 
50  35  11  77  369  0 
37  22  7  64  49 
34  10  76  61  360  9 
369  369  369 
369  369  369 
369  369  369 
369  

 Adjust the values in the center row by adding and subtracting the values in the last columns to generate 4.
At this point fourduplicates (in light orange) have been generated.
 Generate a mask using n^{2} as a factor to be added to the appropriate cell of square 5. Since two duplicates are present in
columns 4 and 6, two n^{2} factors must be present per row, column and diagonal. In this case to the center cell is added
2n^{2}.
4
 369 
21  6  72  48  24  18 
75  60  45  369 
5  71  47  32  17 
74  59  44  20  369 
70  46  31  16  82 
58  43  19  4  369 
54  30  15  81  48 
42  27  3  69  369 
29  14  80  56  41 
26  2  68  53  369 
13  79  55  40  34 
1  67  52  28  369 
78  63  39  24  0 
66  51  36  12  369 
62  38  23  8  65 
50  35  11  77  369 
37  22  7  64  58 
34  10  76  61  369 
369  369  369 
369  369  369 
369  369  369 
369 

+ 
Mask B
   81  
  81  
81   81   
   
    
81   81  
81     
 81   
    162 
   
  81   
   81 
 81   81  
   
    
 81   81 
 81    
81    

⇒ 
 Addition of these factors to square 4 gives square 85 with S = 531 = ½(n^{3} + 37n) and the
modification of 17 numbers.
5
 531 
21  6  72  129  24  18 
75  141  45  531 
86  71  128  32  17 
74  59  44  20  531 
70  46  31  16  82 
139  43  100  4  531 
135  30  15  81  48 
42  108  3  69  531 
29  14  80  56  203 
26  2  68  53  531 
13  79  136  40  34 
1  67  52  109  531 
78  144  39  105  0 
66  51  36  12  531 
62  38  23  8  65 
50  116  11  158  531 
37  103  7  64  58 
115  10  76  61  531 
531  531  531 
531  531  531 
531  531  531 
531 
This completes this section on a new slant break Loubère Type 5x5 and 9x9 MaskGenerated Methods. To return to homepage.
Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com