A New Procedure for Magic Squares (Part II)
Consecutive Internal 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 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 9x9 Magic Square
Method: Reading boustrophedonically (like a sidewinder snake)  use of mask
 Construct the 9x9 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 36 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 45 along the
khaki path(Square 2).
1
 1   2   3 
 4  
5   6   7 
 8   9 
 10   11  
12   13  
18   17   16 
 15   14 
    
   
19   20   21 
 22   23 
 27   26  
25   24  
28   29   30 
 31   32 
 36   35  
34   33  

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

⇒ 
 On reaching 63 fill in 64 (ast cell in the first row).
 Fill in the first row then add 69 to the khaki cell on the second row. This is the only row that has its entries added
in reverse to the other rows as was mentioned above.
 Since the columns are all equal to 369 add or subtract the numbers in the last row from the center column values to generate square 4.
At this point four duplicates have been generated.
3
 546 
68  1  67  2  66  3 
65  4  64  340  29 
5  72  6  71  7 
70  8  69  9  317  52 
77  10  76  11  75 
12  74  13  73  421  52 
18  78  17  79  16 
80  15  81  14  398  29 
37  44  39  42  41 
40  43  38  45  369  0 
19  46  20  47  21 
48  22  49  23  295  74 
54  27  53  26  52 
25  51  24  50  362  7 
28  55  29  56  30 
57  31  58  32  376  7 
63  36  62  35  61 
34  60  33  59  443  74 
369  369  369 
369  369  369 
369  369  369 
552  

⇒ 
4
 546 
68  1  67  2  95  3 
65  4  64  369 
5  72  6  71  59 
70  8  69  9  369 
77  10  76  11  23 
12  74  13  73  369 
18  78  17  79  13 
80  15  81  14  369 
37  44  39  42  41 
40  43  38  45  369 
19  46  20  47  95 
48  22  49  23  369 
54  27  53  26  59 
25  51  24  50  369 
28  55  29  56  23 
57  31  58  32  369 
63  36  62  35  13 
34  60  33  59  369 
369  369  369 
369  369  369 
369  369  369 
552 

 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 the magic sum.
 We start by subtracting 369 from each of the diagonals(546,552) (square 4) to give 177 and 183, respectively and which will be used as what I call the
"de la Hire constants".
Addition of 177 and or 183 to the diagonals and columns and rows gives 729 a magic presum.:
The right diagonal: 729 = 546 + 183
The left diagonal: 739 = 552 + 177
Columns and rows: 729 = 369 +177 + 183
 Since four numbers from the center column must be modified the equations are modified such that the following conditions are obeyed:
The right diagonal: 1089 = 546 + 177 + 2(183)
The left diagonal: 1089 = 546 + 2(177) + 183
The rows and columns: 1089 = 369 + 2(177) + 2(183).
 Generate the mask using the 177 and 183 factors adding these factors to the appropriate cells in square 4 to generate square 5.
 Square 5 has a magic sum equal to 1089, i.e., S = 1089 = ½(n^{3} + 161n).
+ 
Mask A
 183   177  177  
183   
177   177  183   
 183  
183  177    183  
 177  
   177   183 
183   177 
177   177    
 183  183 
     183 
177  177  183 
 183   183  177 
177    
183  177  183   
177    
  183   183  
177   177 

⇒ 
5
 1089 
68  184  67  179  272  3 
248  4  64  1089 
182  72  183  254  59 
70  8  252  9  1089 
260  187  76  11  206 
12  74  190  73  1089 
18  78  17  256  13 
263  198  81  191 
1089 
214  44  216  42  41 
40  43  221  228  1089 
19  46  20  47  95 
231  199  226 
206  1089 
54  210  53  209  236 
202  51  24  50  1089 
211  232  212  56  23 
234  31  58  32  1089 
63  36  245  35  170 
34  237  33  236  1089 
1089  1089  1089 
1089  1089  1089 
1089  1089  1089 
1089 

This completes this section on a new Consecutive Internal 9x9 MaskGenerated Squares (Part II). The next section deals with
Consecutive Internal 13x13 MaskGenerated Squares (Part III). To return to homepage.
Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com