New De La Loubère Method and Squares (Part IIIA)

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

the Full Monty III

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

This page which is a continuation of new Loubère Knight methods takes this new approach further. this site will show that the Loubère-knight method can use knight moves of variable lengths, for example all nxn squares within the group may use the same knight move. Previously the knight move was 2 down, 1 left for Loubère for each nxn square and variable for each Bachet de Méziriac square. Only several variable knight moves appear to work and I will only cover three of these. To start we place the 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[2,1] (2 down,1 left) for the broken yellow diagonal and break/knight[1,2] (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 + 7).

the set of Broken Diagonals
11
11
11
1 1
11
11
11
 
center Value
EquationValue K[4D,3L]EquationValue K[2U,3R]
½(n2 + 5)27½(n2 - 3)23
½(n2 + 7)28½(n2 - 1)24
½(n2 - 5)22½(n2 + 1)25
½(n2 - 3)23½(n2 + 3)26
½(n2 - 1)24½(n2 + 5)27
½(n2 + 1)25½(n2 + 7)28
½(n2 + 3)26½(n2 - 5)22

These new Loubère squares, which I will label Ln* (center cell#) (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 on this page exhibit the following properties:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For the Knight[4D,3L] method no odd squares divisible by 3 or 5 are magic.
  3. For the Knight[2U,3R] method no odd squares divisible by 3 are magic.

Construction of Regular and Non-regular Loubère-Knight squares

Examples of Magic 7x7 Squares Using the Knight[4D,3L] Method

  1. Place the number 1 at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Using a knight move, move 4 down and 3 left.
  3. Repeat the process until the square is filled, as shown below in squares 1-4 (Show only first knight move per square).
1
  1 13
7 12
611
510
9 4
3 8
  2 14
2
1 13 18 23
7 12 17 22
611 16 28
51015 27
9 21 26 4
20 25 3 8
24 2 14 19
3
35 40 45 1 13 18 23
39447 12 17 22 34
43611 16 28 33 38
51015 27 32 37 49
9 21 26 31 36 48 4
20 25 30 42 47 3 8
24 29 41 46 2 14 19
4 L7* 27 K[4D,3L]
35 40 45 1 13 18 23
39447 12 17 22 34
43611 16 28 33 38
51015 27 32 37 49
9 21 26 31 36 48 4
20 25 30 42 47 3 8
24 29 41 46 2 14 19
A seven series

Note that the square is non-regular, i.e, not complementary as shown in the connectivities of the complementary table. Also to decrease crowding only the first eight entries of the connectivity table are shown.

Three 7x7 Knight[4D,3L] Examples

Three examples showing the complete square and the first knight break:

1 L7* 22 K[4D,3L]
30 42 47 3 8 20 25
41462 14 19 24 39
45113 18 23 35 40
71217 22 34 39 44
11 16 28 33 38 43 6
15 27 32 37 49 5 10
26 31 36 48 4 9 21
 
2 L7* 24 K[4D,3L]
32 37 49 5 10 15 27
36484 9 21 26 31
4738 20 25 30 42
21419 24 29 41 46
13 18 23 35 40 45 1
17 22 34 39 44 7 12
28 33 38 43 6 11 16
 
3 L7* 26 K[4D,3L]
34 39 44 7 12 17 22
38436 11 16 28 33
49510 15 27 32 37
4921 26 31 36 48
8 20 25 30 32 47 3
19 24 29 41 46 2 14
23 35 40 45 1 13 18

Examples of Magic 7x7 Squares Using the Knight[2U,3R] Method

  1. Place the number 1 at the center of the first row of a 7x7 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 up and 3 right.
  3. Repeat the process until the square is filled, as shown below in squares 1-4 (Show only first knight move per square).
 
1
1 14
7 13
612
511
10 4
3 9
2 8
2
1 14 20 26
7 13 19 25
612 18 24
51117 23
10 16 22 4
15 28 3 9
27 2 8 21
3
32 38 44 1 14 20 26
37437 13 19 25 31
49612 18 24 30 36
51117 23 29 42 48
10 16 22 35 41 47 4
15 28 34 40 46 3 9
27 33 39 45 2 8 21
4 L7* 23 K[2U,3R]
32 38 44 1 14 20 26
37437 13 19 25 31
49612 18 24 30 36
51117 23 29 42 48
10 16 22 35 41 47 4
15 28 34 40 46 3 9
27 33 39 45 2 8 21
A seven series

Note that the square is non-regular, i.e, not complementary as shown in the connectivities of the complementary table. Also to decrease crowding only the first six entries of the connectivity table are shown.

Three 7x7 Knight[2U,3R] Examples

Three examples showing the complete square and the first knight break:

1 L7* 25 K[2U,3R]
34 40 46 3 9 15 28
39452 8 21 27 33
44114 20 26 32 38
71319 25 31 37 43
12 18 24 30 36 49 6
17 23 29 42 48 5 11
22 35 41 47 4 10 16
 
2 L7* 27 K[2U,3R]
29 42 48 5 11 17 23
41474 10 16 22 35
4639 15 28 34 40
2821 27 33 39 45
14 20 26 32 38 44 1
19 25 31 37 43 7 13
24 30 36 49 6 12 18
 
3 L7* 28 K[2U,3R]
30 36 49 6 12 18 24
42485 11 17 23 29
47410 16 22 35 41
3915 28 34 40 46
8 21 27 33 39 45 2
20 26 32 38 44 1 14
25 31 37 43 7 13 19

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* 25 K[2U,3R] one can move up the right diagonal on a plane of four L7* 28 K[2U,3R] 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 this series go to New De La Loubère Method and Squares (Part IIIB). To return to homepage.


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