New De La Loubère Two Step Staircase and Knight Methods and Squares (Part I)
Regular and NonRegular Squares
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 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 

Loubère squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time.
In this new method each subsequent number is added in a stepwise two step manner.
In addition, n odd squares may be constructed with the initial number 1 on either of two broken diagonals shown below in
light blue or yellow separated by symmetry by a
light grey diagonal for a 5x5 square.
>The set of 5x5 Loubère Broken Diagonals
 1 
1 
 
1  1  
 
1   
 1 
  
1  1 
  1 
1  


The set of 7x7 Loubère Broken Diagonals
 
1  1 
 

 1  1 
 
 
1  1 
 

 
1   
 
 1 
 
 
 1  1 
 
 
1  1  
 
 1  1 
 

These new Loubère squares, which I will label Ln^{*} (center cell#) [2S,D or L] where
Ln^{*} signifies a two step nxn
Loubère square with the center cell number of the square and breaking either down or to
the left.
In the second method
the squares are labeled LKn^{*} (center cell#) [2S,(2D,1L)] where Ln^{*} signifies a
two step nxn Loubère square with the center cell number of the square and breaking either in knight fashion (2 down,1 left) for
yellow and
(1 down,2 left) for light blue.
 Every number on the main diagonal is represented at least once in this type of square.
 For Loubère odd squares where n is divisible by 3, i.e., 3(2n + 1) are nonmagic,
while odd squares where n is divisible by 5 are semimagic. All others are magic.
 For Loubère Knight squares odd squares n is divisible by 3 are nonmagic and those where n is divisible by 7 are semimagic.
All others are magic.
Construction of the Regular 5x5 Two Step Loubère Magic Square
5x5 Squares
 To generate the regular square, L5^{*} 13 [2S,D],
place a 1 into the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right in steps of 2 until blocked by a previous number.
 Move one cell down.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

⇒ 
3
  1 
10  14 
 3  7 
11  
5  9  13 
17  
6  15  
 2 
12  16  
4  8 

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

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

The Four Nonregular 5x5 Two Step Loubère Semimagic Squares
The fully constructed nonregular squares shown below have left diagonal S values of 55, 70, 60 and 75, respectively.
A L5^{*} 11 [2S,D]
16  25  4 
8  12 
22  1  10 
14  18 
3  7  11 
20  24 
9  13  17 
21  5 
15  19  23 
2  6 


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


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


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

The Regular 7x7 Two Step Loubère Square
The construction of the 7x7 regular Loubère square L7^{*} 25 [2S,D] from the
broken yellow diagonal is shown below again in steps of 2.

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

⇒ 
2
  
1  13 
18  23 
  4 
9  21 
26  
 7  12 
17  22 
 
3  8  20 
25  30 
 
11  16  28 
 
 6 
19  24  29 
 
2  14 
27   
 5 
10  15 

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

Construction of a Nonregular 5x5 Two step Loubère Knight Magic Square
5x5 Squares
 To generate the nonregular square, LK5^{*} 13 [2S,(2D,1L)],
place a 1 into the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right in steps of 2 until blocked by a previous number.
 Move two cells down one cell left (the knight break).
 Repeat the process until the square is filled, as shown below in squares 15.

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

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

⇒ 
4 LK5^{*} 14 [2S,(2D,1L)]
17  24  1 
8  15 
21  3  10 
12  19 
5  7  14 
16  23 
9  11  18 
25  2 
13  20  22 
4  6 

The Other Four Regular and Nonregular 5x5 Two Step Loubère Knight magic Squares
A LK5^{*} 12 [2S,(2D,1L)]
20  22  4 
6  13 
24  1  8 
15  17 
3  10  12 
19  21 
7  14  16 
23  5 
11  18  25 
2  9 


B LK5^{*} 15 [2S,(2D,1L)]
18  25  2 
9  11 
22  4  6 
13  20 
1  8  15 
17  24 
10  12  19 
21  3 
14  16  23 
5  7 


C LK5^{*} 13 [2S,(2D,1L)]
16  23  5 
7  14 
25  2  9 
11  18 
4  6  13 
20  22 
8  15  17 
24  1 
12  19  21 
3  10 


D LK5^{*} 11 [2S,(2D,1L)]
19  21  3 
10  12 
23  5  7 
14  16 
2  9  11 
18  25 
6  13  20 
22  4 
15  17  24 
1  8 

A NonRegular 7x7 Two Step Loubère Knight Square
The construction of a 7x7 nonregular two Step Loubère Knight square LK7^{*} 23 [2S,(2D,1L)]
from the broken yellow diagonal similar to the method above is shown below again in steps of 2, followed by a knight break.
The last table shows also the sums (S) of the left main diagonal for all the 7x7 regular and nonregular squares in this set.
1
  
1  10 
 
  4 
13  15 
 
 7  9 
 
 
3  12  
 
 
8   
 
 6 
  
 
2  11 
  
 5 
14  

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

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


7x7 Cell Values and S of K[2S,(2D,1L)]
Center Value  S + d  d 
23  161  14 
27  189  14 
24  168  7 
28  196  21 
25  175  0 
22  154  21 
26  182  7 

The Plane of Loubère 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
LK7^{*} 23 [2S,(2D,1L)] one can move up the right diagonal on a plane of four
LK7^{*} 23 [2S,(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 De La Loubère two step squares (Part I). The next section deals with a
new Méziriac 3 step method (Part II). To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com