A New Procedure for Magic Squares (Part III)
Consecutive 9x9 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 leftmost column and entered across every other cell. Consecutive numbers are then added
to the next rows boustrophedonically or in regular left to right order. 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 a modified sum equation shown in
New block Loubère Method):
S = ½(n^{3} ± an)
Construction of a 9x9 Magic Square
Method: Reading from left to right  use of mask
 Construct Square 1 by adding consecutive numbers numbers to every other cells. Do not fill in the center row at this moment.
 After reaching number 36 jump to the center row and fill it consecutively (Square 2).
 After reaching 45 go down one cell and continue filling the cells up to 59 (Square 3). At this point fill in in the number 60 into the second cell of the
last row and continue to the numeber 63.
1
1   2   3  
4   5 
 6   7  
8   9  
10   11   12 
 13   14 
 15   16  
17   18  
    
   
 22   21  
20   19  
27   26   25 
 24   23 
 31   30  
29   28  
32   33   34 
 35   36 

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

⇒ 
3
1   2   3  
4   5 
 6   7  
8   9  
10   11   12 
 13   14 
 15   16  
17   18  
37  38  39  40  41 
42  43  44  45 
50  22  49  21  48 
20  47  19  46 
27  54  26  53  25 
52  24  51  23 
59  31  58  30  57 
29  56  28  55 
32  60  33  61  34 
62  35  63  36 

 From 63 go up the column and insert 64 into the cell and continue filling in a reverse manner on all of the four top rows.
 At this point not all columns, rows or diagonals sum to 369.
 Where the grey sums on the next to the last right hand column right intersect the grey sums in the next to the last row, adjust the values in these cells by
adding and subtracting the values in the last row and columns to generate 5. At this point five duplicates have been generated.
4
 195  
1  67  2  66  3  65 
4  64  5  277  92 
72  6  71  7  70 
8  69  9  68  380  11 
10  76  11  75  12 
74  13  73  14  358  11 
81  15  80  16  79 
17  78  18  77  461  92 
37  38  39  40  41 
42  43  44  45  369  0 
50  22  49  21  48 
20  47  19  46  322  47 
27  54  26  53  25 
52  24  51  23  335  34 
59  31  58  30  57 
29  56  28  55  403  34 
32  60  33  61  34 
62  35  63  36  416  47 
369  369  369 
369  369  369 
369  369  369 
183  

⇒ 
5
 195 
1  67  2  66  95  65 
4  64  5  369 
72  6  71  7  59 
8  69  9  68  369 
10  76  11  75  23 
74  13  73  14  369 
81  15  80  16  13 
17  78  18  77  369 
37  38  39  40  41 
42  43  44  45  369 
50  22  49  21  95 
20  47  19  46  369 
27  54  26  53  59 
52  24  51  23  369 
59  31  58  30  23 
29  56  28  55  369 
32  60  33  61  13 
62  35  63  36  369 
369  369  369 
369  369  369 
369  369  369 
183 

+ 
 A possible mask for for converting square 5 into square 6 adds either 174 or 186 of mask A to the appropriate cells of square 5 (see box below).
 We start by subtracting each of the diagonals(183,195) from square 7 from 369 to give 186 and 174, respectively and which will be used as what I call the
"de la Hire constants".
Addition of the sum of these two numbers, 186 + 174 = 360 to
369 gives 729 a magic presum.
 If we subtract the sum 775 from the two diagonals we obtain the following two sums:
729 = 183 + 546 and 729 = 195 + 534.
 The following solutions are obtained:
The left diagonal: 729 = 183 + 546 = 183 + 174 + 2(186)
The right diagonal: 729 = 195 + 534 = 195 + 2(174) + 186
The rows and columns: 729 = 369 + 174 + 186.
 However, four cells in the center column of square 5 must be changed for all duplicates to disappear.
In order for no duplicates to occur the equations must be changed such that S = 1089 and the equations change accordingly:
The left diagonal: 1089 = 183 + 906 = 183 + 2(174) + 3(186)
The right diagonal: 1089 = 195 + 894 = 195 + 3(174) + 2(186)
The rows and columns: 1089 = 369 + 720 = 369 + 2(174) + 2(186).
 Generate the mask using the 174 and 186 factors and adding these factors to the appropriate cells in square 5 to generate square 6. Sometimes duplicates
are formed so go back and tweak the mask a little (move some of the cell numbers around) until no duplicates are generated.
 Square 6 has a magic sum equal to 1089, i.e., S = 1089 = ½(n^{3} + 161n).
Mask A
174    
174   186   186 
 186   174  186 
  174  
  186   
186  174   174 
174  186  174   
 186   
 174   186  
186   174  
 174   186  
 174  186  
186   174   
174    186 
186    174  174 
  186  
  186   186 
174    174 

⇒ 
6
 1089 
175  67  2  66  269  65 
190  64  191  1089 
72  192  71  181  245 
8  69  183  68  1089 
10  76  197  75  23 
260  187  73  188 
1089 
255  201  254  16  13 
17  264  18  77  1089 
37  212  39  226  41 
228  43  218  45  1089 
50  196  49  207  95 
20  221  205  46  1089 
213  54  200  53  59 
226  24  51  209  1089 
245  31  58  204  197 
29  56  214  55  1089 
32  60  219  61  173 
236  35  63  210  1089 
1089  1089  1089 
1089  1089  1089 
1089  1089  1089 
1089 

This completes this section on a consecutive 9x9 MaskGenerated Methods (Part III). Next section deals with consecutive a 9x9 MaskGenerated Boustrophedonic Method
(reading right then left) Part IV.
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Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com