New De La Loubère Method and Squares (Part II)

Regular and Non-Regular Loubère-Knight Combo Magic and Semi-Magic Squares

the Full Monty II

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.
  3. 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
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

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 non-regular 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 non-regular 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(n2 - n + 2) to ½(n2 + 5) for squares having knight[2,1] and knight[1,2] moves.

The set of Broken Diagonals
1 1
11
1 1
1 1
1 1
 
center Value
EquationValue K[1,2]EquationValue K[2,1]
½(n2 - 1)12½(n2 + 5)15
½(n2 - 3)11½(n2 - 3)11
½(n2 + 5)15½(n2 - 1)12
½(n2 + 3)14½(n2 + 1)13
½(n2 + 1)13½(n2 + 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:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. Odd squares divisible by 3, i.e., 3(2n + 1) obey a simple modified rule. these square may be magic or semi-magic 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.
  3. 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 semi-magic squares.

Construction of Regular and Non-regular Loubère-Knight squares

The 5x5 Squares

  1. 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.
  2. Using a knight move, move 2 down and 1 left.
  3. Repeat the process until the square is filled, as shown below in squares 1-4.
1
  1
58
47
6 3
  2
 
2
1 9 12
58 11
4715
614 3
13 16 2 10
3
20 23 1 9 12
2258 11 19
4715 18 21
61417 25 3
13 16 24 2 10
4 L5* 15 K[2D,1L]
20 23 1 9 12
2258 11 19
4715 18 21
61417 25 3
13 16 24 2 10
A five series

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
2319 12 20
5811 19 22
71518 21 4
14 17 25 3 6
 
B
17 25 3 6 14
24210 13 16
1912 20 23
81119 22 5
15 18 21 4 7
 
C
18 21 4 7 15
2536 14 17
21013 16 24
91220 23 1
11 19 22 5 8
D
19 22 5 8 11
2147 15 18
3614 17 25
101316 24 2
12 20 23 1 9

Two Examples of Magic 7x7 Squares

The 7x7 Loubère-Knight square method may also be used to construct the following two non-regular 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
86
5 14
15 13 4
12 3
11 2
10 1
2
18 9 7 29 27
86 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
8646 37 35 26 17
54536 34 25 16 14
444233 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
8646 37 35 26 17
54536 34 25 16 14
444233 24 15 13 4
41 32 23 21 12 3 43
31 22 20 11 2 49 40
2819 10 148 3930
A seven series

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 Loubère-Knight Squares

The next squares exhibiting these pentad properties are those 25x25 magic and semi-magic 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 Sleft 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 Sleft+d2Value d2
3023073123173227800-n
30330831331832378250
3043093143193247850n
30531031532032578752n
3063113163213017775-2n

These may be compared to the table shown in Méziriac type Squares Part IV.

The Plane of Loubère-Knight 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 non-regular Loubère-Knight squares. To continue to Part IIIA. To return to homepage.


Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com