New De La Loubère Method and Squares (Part II)
Regular and NonRegular LoubèreKnight Combo Magic and SemiMagic Squares
the Full Monty II
A Discussion of the New Methods
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 same regular square is produced when the initial 1 is placed on the center of the first row or the center of the last column, however, this is not the case for
the first column or last row.
the 5x5 and 7x7 regular Loubère squares are shown below as examples:
17  24  1 
8  15 
23  5  7 
14  16 
4  6  13 
20  22 
10  12  19 
21  3 
11  18  25 
2  9 

 
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 

Because of the way the regular Loubère square is constructed (placing the starting number 1, in the center of the top row or in the center of the last column)
this was taken as gospel. However, two methods apply. Placing the starting number 1 on either of 2 broken diagonals, filling the square in a Loubère fashion
until a block is encountered, then proceeding with a knight move results in the generation of regular and nonregular squares.
All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the
light grey main diagonal behave differently from typical
Loubère squares. After a break/ knight[2D,1L] (2 Down,1 Left) for the broken yellow diagonal
and break/knight[1D,2L] (1 Down, 2 Left) for the broken light blue diagonal, one regular square and
n  1
nonregular squares are produced. Squares belonging to one diagonal group are identical to another square on the
other diagonal group.
Using the 5x5 square as an example shows the two diagonals and typical 1 positions. the second table shows the equation and value of the center cell of each square
(starting with the square generated from 1 in the first row) where the values range from
½(n^{2}  n + 2) to ½(n^{2} + 5) for squares
having knight[2,1] and knight[1,2] moves.
The set of Broken Diagonals
 1 
1 
 
1  1  
 
1   
 1 
  
1  1 
  1 
1  


center Value
Equation  Value K[1,2]  Equation  Value K[2,1] 
½(n^{2}  1)  12  ½(n^{2} + 5)  15 
½(n^{2}  3)  11  ½(n^{2}  3)  11 
½(n^{2} + 5)  15  ½(n^{2}  1)  12 
½(n^{2} + 3)  14  ½(n^{2} + 1)  13 
½(n^{2} + 1)  13  ½(n^{2} + 3)  14 

these new Loubère squares, which I will label
Ln^{*} (center cell#) (K[L2,L1] or K[L1,L2]) where (Ln^{*} signifies a nxn Loubère
square with the center cell number of the square and having a break followed by a Knight move L2= 2 Down L1= 1 Left or a Knight move L1= 1 Down L2= 2 Left.
(K[L1,L2] or K[L3,L4]) where (Ln^{*} signifies a nxn
Loubère square with the center cell number of the square and having a break followed by a Knight move L1= number1(Up) and
L2=number2(Right) Or a Knight move L3= number3(Down) and L4=number4(Left).
this is opposed to the original Loubère squares depicted in the introduction as
L5^{*} 13 [1D] and L7^{*} 25 [1D].
the squares exhibit the following properties:
 Every number on the main diagonal is represented at least once in this type of square.
 Odd squares divisible by 3, i.e., 3(2n + 1) obey a simple modified rule. these square
may be magic or
semimagic and the sum of the left main diagonal cycles through the triad S, S + n and S  n.
For example, for a 9x9 square there are three cycles of 369, 370 and 369.
 Odd squares divisible by 5 but not 3 produce a pentad of squares having the sums S  2n, S  n,
S, S + n and S + 2n. These are repeated n/5 times. For example the 5x5 squares are repeated once,
while the 25x25 are repeated 5 times. the table at the bottom of this page shows the results of all 25 magic and semimagic squares.
Construction of Regular and Nonregular LoubèreKnight squares
The 5x5 Squares
 Place the number 1 at the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked
by a previous number.
 Using a knight move, move 2 down and 1 left.
 Repeat the process until the square is filled, as shown below in squares 14.


2
  1 
9  12 
 5  8 
11  
4  7  15 
 
6  14  
 3 
13  16  
2  10 

⇒ 
3
20  23  1 
9  12 
22  5  8 
11  19 
4  7  15 
18  21 
6  14  17 
25  3 
13  16  24 
2  10 

⇒ 
4 L5^{*} 15 K[2D,1L]
20  23  1 
9  12 
22  5  8 
11  19 
4  7  15 
18  21 
6  14  17 
25  3 
13  16  24 
2  10 

Note that two pairs sum to 30 including the center square when multiplied by 2 and ten pairs sum to 25, plus the main left diagonal sums to 75.
In addition, none of these pairs are complementary, as in the regular Loubère squares, as shown in the connectivities of the complementary table.
the following 4 squares complete the pentad of 5x5 squares. the sums of the left diagonal are 55, 60, 65 and 70, respectively.
A
16  24  2 
10  3 
23  1  9 
12  20 
5  8  11 
19  22 
7  15  18 
21  4 
14  17  25 
3  6 


B
17  25  3 
6  14 
24  2  10 
13  16 
1  9  12 
20  23 
8  11  19 
22  5 
15  18  21 
4  7 


C
18  21  4 
7  15 
25  3  6 
14  17 
2  10  13 
16  24 
9  12  20 
23  1 
11  19  22 
5  8 

⇒ 
D
19  22  5 
8  11 
21  4  7 
15  18 
3  6  14 
17  25 
10  13  16 
24  2 
12  20  23 
1  9 

Two Examples of Magic 7x7 Squares
The 7x7 LoubèreKnight square method may also be used to construct the following two nonregular magic nxn squares such as the 7x7 picked
from each of the broken diagonal and are shown below with their corresponding complementary tables.
1
 9  7 
 
 
8  6  
 
 
5   
 
 14 
  
 15 
13  4 
  
 12 
3  
  
11  2 
 
  10 
1  
 

⇒ 
2
18  9  7 
 
29  27 
8  6  
 
26  17 
5   
 25 
16  14 
  
24  15 
13  4 
  23 
21  12 
3  
 22  20 
11  2 
 
28  19  10 
1  
 

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

⇒ 
4 L7^{*} 24 K[1D,2L]
18  9  7 
47  38 
29  27 
8  6  46 
37  35 
26  17 
5  45  36 
34  25 
16  14 
44  42  33 
24  15 
13  4 
41  32  23 
21  12 
3  43 
31  22  20 
11  2 
49  40 
28  19  10 
1  48 
39  30 

Note that seven pairs sum to 55 and seventeen pairs sum to 48. including the center square when multiplied by 2.
In addition none of these pairs are complementary, as in the regular Loubère squares, as shown in the connectivities of the complementary table.
Results of 25x25 LoubèreKnight Squares
The next squares exhibiting these pentad properties are those 25x25 magic and semimagic squares that break using the K[2D,1L] knight
moves. If we tabulate the center numbers of each of the 25x25 squares we get the table, where S_{left} is the sum of the main
left diagonal and the yellow entry signifies that all the squares having those centers on that line are magic:
25x25 Square Values
center of Square  S_{left}+d2  Value d2 
302  307  312  317  322  7800  n 
303  308  313  318  323  7825  0 
304  309  314  319  324  7850  n 
305  310  315  320  325  7875  2n 
306  311  316  321  301  7775  2n 
These may be compared to the table shown in Méziriac type Squares Part IV.
The Plane of LoubèreKnight Squares
At this point it may be said that alternatively these squares may be constructed using a plane of four squares. For example using the 7x7 square
L7^{*} 28 K[2D,1L] one can move up the right diagonal on a plane
of four L7^{*} 28 K[2D,1L] and generate
the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.
this completes this section on regular and nonregular LoubèreKnight squares. To continue to Part IIIA.
To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com