A New Procedure for Magic Squares (Part VII) Continuation
Loubère Knight Block transModified 7x7 Squares
A Discussion of the New Method
An important general principle for generating odd magic squares by the De La Loubère method is that the center cell must always contain the middle number of
the series of numbers used, 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.
The 7x7 regular Loubère square is shown below:
30  39  48 
1  10 
19  28 
38  47  7 
9  18 
27  29 
46  6  8 
17  26 
35  37 
5  14  16 
25  34 
36  45 
13  15  24 
33  42 
44  4 
21  23  32 
41  43 
3  12 
22  31  40 
49  2 
11  20 
Normally the Loubère and Méziriac methods involve a stepwise approach of consecutive numbers, i.e., 1,2,3...n^{2}.
For example, A new generalized Loubère procedure where the initial number is placed anywhere on any
n  1 cells of the first row.
However, in this new method although the numbers are added consecutively starting with 1 and ending with
n^{2}, which are then used to generate modified
Loubère having numbers greater than n^{2}.
The nonmagic square may use a variety of methods. For example, adding a constant to a
nonmagic diagonal and converting that diagonal into one that is readily amenable
to a transposition (trans) is one method.
Accordingly, the numbers are added consecutively starting out with 1 and ending with numbers greater than n^{2}. Whenever a break
occurs a knight move is performed instead of a normal translational move (up, down or sideways). The knight move is either a (2,1) or a variable knight move , e.g (3,2),
for Méziriac squares squares or Loubère squares. Both these square types were covered in Loubère Knight and
Méziriac Knight square methods. Dual modification is then performed as discussed above.
Moreover, it must be stated here that the magic sum has been modified from the known equation
S = ½(n^{3} + n) to
the general equation:
S = ½(n^{3} ± an)
which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or 
values. For example when a = 1 the normal magic sum S is implied.
When a takes on different odd values S gives the magic sum of a modified magic square.
It will be shown that the addition or subtraction of n^{2} to some of the cells in
the square gives rise to a new magic square.
transConversion of Loubère nonmagic Squares to
Magic Squares
Method I: Start on the first row center (1 ⇒ (2,1) up knight break)
 Begin generating the nonmagic Loubère square by placing 1 in the center cell of the first row and add six nonconsecutive
using heptad table I and method of New generalized Loubère procedure.
 Perform a knight break two cells right one cell up as shown in color
(this is done in order to produce a nonmagic square).
 Repeat the process until the square is filled as shown below in squares 12.
 As shown square 2 is nonmagic and the last column corresponds to how much must be added or subtracted to the previous column numbers to get to 175.

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

⇒ 
 The next series of moves (in groups of 2), adding or subtracting numbers from a row converts all the sums to 175 with the generation of duplicate numbers.
 At this point (square 3) four numbers are duplicates (in blue
and orange ).
 In order to remove these duplicates add n^{2} = 49 to each row so that each row, column and diagonal have
exactly one modification.
 Square 4 is produced whereby all the sums have been converted to the magic sum 224, and
S = ½(n^{3} + 15n).
3
 175 
29  36  64  1 
8  15  22  175 
38  45  3  17 
17  24  31  175 
47  5  12  12 
26  33  40  175 
2  9  16  23 
30  51  44  175 
13  20  27  34 
41  34  6  175 
21  28  14  42 
49  7  14  175 
25  32  39  46 
4  11  18  175 
175  175  175 
175  175  175 
175  175 

⇒ 
4
 224 
29  85  64  1 
8  15  22  224 
38  45  3  66 
17  24  31  224 
47  5  61  12 
26  33  40  224 
2  9  16  23 
79  51  44  224 
13  20  27  34 
41  83  6  224 
21  28  14  42 
49  7  63  224 
74  32  39  46 
4  11  18  224 
224  224  224 
224  224  224 
224  224 

Method II: Start at first row center (1 ⇒ (2,1) down knight break)
A double Modification
 Begin generating the nonmagic Loubère square by placing 1 in the center cell of the first row and
performing a knight break move down.

 Fill the square as shown to produce nonmagic square 2.
 As shown square 2 the last row corresponds to how much must be added or subtracted to the previous column numbers to get to 182 (the sums in the last column.
1
   1 
  4 
   
 3  
   
2   
   8 
  11 
  7  
 10  
 6   
9   
5    
  

⇒ 
2
 35 
40  27  14  1 
30  17  4  133 
26  13  49  36 
16  3  39  182 
12  48  35  22 
2  38  25  182 
47  34  21  8 
37  24  11  182 
33  20  7  43 
23  10  46  182 
19  6  42  29 
9  45  32  182 
5  41  28  15 
44  31  18  182 
182  189  196 
154  161  168 
175  182 
0  7  14  +28 
+21  +14  +7  

⇒ 
 To begin converting this square into a magic one, add 28 to each of the entries of the right blue diagonal.
 At this point (square 3) all the sums have changed by 28.
 Transpose (trans), two cells up, all the 7 numbers from the blue diagonal of square 2 to the second adjacent left diagonal (square 4).
At this point the highest sum possible is 231. This sum may change.
3
 231 
40  27  14  1 
30  17  32  161 
26  13  49  36 
16  31  39  210 
12  48  35  22 
30  38  25  210 
47  34  21  36 
37  24  11  210 
33  20  35  43 
23  10  46  210 
19  34  42  29 
9  45  32  210 
33  41  28  15 
44  31  18  210 
210  217  224 
182  189  196 
203  210 
28  21  14  56 
49  42  35  

⇒ 
4
 231 
40  27  14  1 
2  17  32  133 
26  13  49  8 
16  31  39  182 
12  48  7  22 
30  38  25  182 
47  6  21  36 
37  24  11  182 
5  20  35  43 
23  10  46  182 
19  34  42  29 
9  45  4  182 
33  41  28  15 
44  3  18  182 
182  189  196 
154  161  168 
175  182 
0  7  14  +28 
+21  +14  +7  

⇒ 
 The next series of moves (in groups of 2), adding or subtracting numbers from a column converts all the sums to 182 with the generation of duplicate numbers
in green.
 Adding n^{2} = 49 to (26, 41, 7, 23, 38 and 42),
shown in orange , removes four duplicates and gives the semimagic square 6.
 In the second modification adding n^{2} = 49 to (9, 10, 11, 12, 13, 14 and 15)
gives square 7 where the magic sum is 280 and where S = ½(n^{3} + 31n).
5
 231 
40  27  14  29 
23  17  32  182 
26  13  49  8 
16  31  39  182 
12  41  7  22 
30  38  32  182 
47  6  7  36 
37  38  11  182 
5  20  35  43 
23  10  46  182 
19  34  42  29 
9  45  4  182 
33  41  28  15 
44  3  18  182 
182  182  182 
182  182  182 
182  182 

⇒ 
6
 280 
40  27  14  29 
23  17  81  231 
75  13  49  8 
16  31  39  231 
12  41  7  22 
30  87  32  231 
47  6  56  36 
37  38  11  231 
5  20  35  43 
72  10  46  231 
19  34  42  78 
9  45  4  231 
33  90  28  15 
44  3  18  231 
231  231  231 
231  231  231 
231  231 

⇒ 
7
 280 
40  27  63  29 
23  17  81  280 
75  62  49  8 
16  31  39  280 
61  41  7  22 
30  87  32  280 
47  6  56  36 
37  38  60  280 
5  20  35  43 
72  59  46  280 
19  34  42  78 
58  45  4  280 
33  90  28  64 
44  3  18  280 
280  280  280 
280  280  280 
280  280 

This completes this section on the new block Loubère Knight Block transModified 7x7 Squares (Part VII).
To continue to section VIII A New Procedure for Magic Squares Continuation.
To return to homepage.
Copyright © 2009 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com