A New Procedure for Magic Squares (Part IB)
Consecutive Boustrophedonic 7x7 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 first leftmost column and entered across every other cell. Consecutive numbers
are then added to the next rows boustrophedonically. At the end the square is filled in by adding numbers to the second cell on the first row and proceeding
as before. The square which is not magic is modified into a form
which can be converted into a magic one 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 follow the sum equation shown in the
New block Loubère Method.
S = ½(n^{3} ± an)
Construction of a 7x7 Magic Square I
Method: Reading consecutive from left to right boustrophedonically  use of a mask
 Construct the 7x7 Square 1 where 7 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell, then swinging around at
the end (boustrophedonically) until the final cell is reached (square 1).
 On reaching this cell go to cell 6 row 1 (next to last) and continue adding consecutive numbers in reverse.
This method is equivalent to adding the requisite numbers on a line and then using a (1 down, 1 left) break as shown for the 4 → 5 move.
 Since the sums of all the columns or rows are not equal to 175 add or subtract the numbers in the last row to the center row and
add or subtract the lst column to the center column. At this point three duplicates (in pink) have been generated (Square 3).
1
1   2 
 3   4 
 7  
6   5  
8   9 
 10   11 
 14  
13   12  
15   16 
 17   18 
 21  
20   19  
22   23 
 24   25 

⇒ 
2
 91  
1  28  2  27  3  26 
4  91  84 
29  7  30 
6  31  5  32  140  35 
8  35  9 
34  10  33  11  140  35 
36  14  37 
13  38  12  39  189  14 
15  42  16 
41  17  40  18  189  14 
43  21  44 
20  45  19  46  238  63 
22  49  23 
48  24  47  25  238  63 
154  196  161 
189  168  182 
175  91  
21  21  14 
14  7  7  0   

⇒ 
3
 63  
1  28  2  111  3  26 
4  175 
29  7  30 
41  31  5  32  175 
8  35  9 
69  10  33  11  175 
57  7  51 
15  45  5 
39  175 
15  42  16 
27  17  40  18  175 
43  21  44 
22  45  19  46  175 
22  49  23 
15  24  47  25  175 
175  175  175 
175  175  175 
175  63  

+ 
 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 be equal to
a magic sum.
 We start by subtracting the diagonals(63,63) from 175 to give 112 and 112, respectively and which will be used as what I call the
"de la Hire constants".
Addition of the constants to 175, i.e., 175 + 112
gives 287 a magic presum.
 The following equations are used such that
the following conditions are obeyed:
The right diagonal and left diagonals : 287 = 63 + 2(112)
The rows and columns: 287 = 175 + 112
 Generate the mask using the 112 factor and adding this factor to the appropriate cells in square 3 generates square 4.
 The equation for this square is 287 = ½(n^{3} + 33n).
Mask A
112   
   
  
  112  
  
   112 
  
112    
 112  
   
  
 112   
  112 
   

⇒ 
4
 287 
113  28  2  111  3  26 
4  287 
29  7  30 
41  31  117  32  287 
8  35  9 
69  10  33  123  287 
57  7  51 
97  45  5  39  287 
15  154  16 
27  17  40  18  287 
43  21  44 
22  157  19  46  287 
22  49  135 
15  24  47  25  287 
287  287  287 
287  287  287 
287  287 

Construction of a 7x7 Magic Square II
Method: Reading consecutive from left to right boustrophedonically  use of a mask
 Construct the 7x7 Square 1 where 7 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell, then swinging around at
the end (boustrophedonically) until the final cell is reached (square 1).
 On reaching this cell go to cell 2 row 1 (second cell) and continue adding consecutive numbers in reverse.
This method is equivalent to adding the requisite numbers on a line and then using a (1 down, 2 right) break as shown for the 4 → 5 move.
 Since the sums of all the columns or rows are not equal to 175 add or subtract the numbers in the last row to the center row and
add or subtract the lst column to the center column. At this point three duplicates (in pink) have been generated (Square 3).
1
1   2 
 3   4 
 5  
6   7  
8   9 
 10   11 
 12  
13   14  
15   16 
 17   18 
 19  
20   21  
22   23 
 24   25 

⇒ 
2
 91  
1  26  2  27  3  28 
4  91  84 
29  5  30 
6  31  7  32  140  35 
8  33  9 
34  10  35  11  140  35 
36  12  37 
13  38  14  39  189  14 
15  40  16 
41  17  42  18  189  14 
43  19  44 
20  45  21  46  238  63 
22  47  23 
48  24  49  25  238  63 
154  182  161 
189  168  196 
175  91  
21  7  14 
14  7  21  0   

⇒ 
3
 63  
1  26  2  111  3  28 
4  175 
29  5  30 
41  31  7  32  175 
8  33  9 
69  10  35  11  175 
57  5  51 
15  45  7 
39  175 
15  40  16 
27  17  42  18  175 
43  19  44 
43  45  21  46  175 
22  47  23 
15  24  49  25  175 
175  175  175 
175  175  175 
175  63  

+ 
 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 be equal to
a magic sum.
 We start by subtracting the diagonals(63,63) from 175 to give 112 and 112, respectively and which will be used as what I call the
"de la Hire constants".
Addition of the constants to 175, i.e., 175 + 112
gives 287 a magic presum.
 The following equations are used such that
the following conditions are obeyed:
The right diagonal and left diagonals : 287 = 63 + 2(112)
The rows and columns: 287 = 175 + 112
 Generate the mask using the 112 factor and adding this factor to the appropriate cells in square 3 generates square 4.
 The equation for this square is 287 = ½(n^{3} + 33n).
Mask B
      112 
 112   
  
  
  112  
   112 
  
112   
   
  
 112   
  112 
   

⇒ 
4
 287 
1  26  2  111  3  28 
116  287 
29  117  30 
41  31  7  32  287 
8  33  9 
69  10  147  11  287 
57  5  51 
97  45  7  39  287 
127  40  16 
27  17  42  18  287 
43  19  44 
43  157  21  46  287 
22  47  135 
15  24  49  25  287 
287  287  287 
287  287  287 
287  287 

This completes this section on a new Consecutive Boustrophedonic 7x7 MaskGenerated Squares.The next section deals with
Consecutive Boustrophedonic 9x9 MaskGenerated Squares (Part II).To return to homepage.
Copyright © 2010 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com