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

Regular and Non-Regular Squares

A Loubere square

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(n2 + 1). The properties of these regular or associated Loubère squares are:

  1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
  2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n2 + 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
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9
 
30 39 48 1 10 19 28
38477 9 18 27 29
4668 17 26 35 37
51416 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
11
1 1
1 1
1 1
 
The set of 7x7 Loubère Broken Diagonals
11
11
11
1 1
11
11
11

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.

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For Loubère odd squares where n is divisible by 3, i.e., 3(2n + 1) are non-magic, while odd squares where n is divisible by 5 are semi-magic. All others are magic.
  3. For Loubère Knight squares odd squares n is divisible by 3 are non-magic and those where n is divisible by 7 are semi-magic. All others are magic.

Construction of the Regular 5x5 Two Step Loubère Magic Square

5x5 Squares

  1. 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.
  2. Move one cell down.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
3
 
2
 
2
1
37
5
6 2
4 8
3
1 10 14
37 11
5913 17
615 2
12 16 4 8
4
18 1 10 14
2437 11 20
5913 17 21
61519 2
12 16 4 8
5 L5* 13 [2S,D]
18 22 1 10 14
2437 11 20
5913 17 21
61519 23 2
12 16 25 4 8

The Four Non-regular 5x5 Two Step Loubère Semi-magic Squares

The fully constructed non-regular 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
22110 14 18
3711 20 24
91317 21 5
15 19 23 2 6
 
B L5* 14 [2S,D]
19 23 2 6 15
2548 12 16
11014 18 22
71120 24 3
13 17 21 5 9
 
C L5* 12 [2S,D]
17 21 5 9 13
2326 15 19
4812 16 25
101418 22 1
11 20 24 3 7
 
D L5* 13 [2S,D]
20 24 3 7 11
2159 13 17
2615 19 23
81216 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
712
38
11 6
2 14
5 10 15
2
1 13 18 23
4 9 21 26
712 17 22
3820 25 30
11 16 28 6
19 24 29 2 14
27 5 10 15
3
35 40 45 1 13 18 23
36484 9 21 26 31
44712 17 22 34 39
3820 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 Non-regular 5x5 Two step Loubère Knight Magic Square

5x5 Squares

  1. To generate the non-regular 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.
  2. Move two cells down one cell left (the knight break).
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
3
5
2
4 6
2
1 8 15
310 12
5714
911 2
13 4 6
3
17 1 8 15
310 12 19
5714 16
91118 2
13 20 4 6
4 LK5* 14 [2S,(2D,1L)]
17 24 1 8 15
21310 12 19
5714 16 23
91118 25 2
13 20 22 4 6

The Other Four Regular and Non-regular 5x5 Two Step Loubère Knight magic Squares

A LK5* 12 [2S,(2D,1L)]
20 22 4 6 13
2418 15 17
31012 19 21
71416 23 5
11 18 25 2 9
 
B LK5* 15 [2S,(2D,1L)]
18 25 2 9 11
2246 13 20
1815 17 24
101219 21 3
14 16 23 5 7
 
C LK5* 13 [2S,(2D,1L)]
16 23 5 7 14
2529 11 18
4613 20 22
81517 24 1
12 19 21 3 10
 
D LK5* 11 [2S,(2D,1L)]
19 21 3 10 12
2357 14 16
2911 18 25
61320 22 4
15 17 24 1 8

A Non-Regular 7x7 Two Step Loubère Knight Square

The construction of a 7x7 non-regular 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 non-regular squares in this set.

1
1 10
4 13 15
79
312
8 6
2 11
5 14
2
1 10 19 28
4 13 15 24
79 18 27 29
31221 23
8 17 26 6
20 22 2 11
25 5 14 16
3
30 39 48 1 10 19 28
42444 13 15 24 33
4779 18 27 29 38
31221 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 ValueS + dd
23161-14
2718914
24168-7
2819621
251750
22154-21
261827

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 non-regular 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. E-Mail: Fiboguti89@Yahoo.com