New De La Loubère Knight-step Method and Squares (Part I)

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 knight step manner, followed by a right or an up break. 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

The new Loubère type squares of Part I, which I will label KLn* (center cell#) [(2D,1L), U or R] where KLn* signifies a Knight-step nxn Loubère square with the center cell number of the square and breaking either to the right or up.

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For Loubère Knight-step where n is divisible by 3, i.e., 3(2n + 1) n squares areproduced where the sum of the left diagonal S equals S + d, and d is equal to either -n2, 0 or n2.

Construction a 5x5 Knight-step Loubère Magic Square

5x5 Squares

  1. To generate the regular square, KL5* 8 [(2D,1L),1R], place a 1 into the center of the first row of a 5x5 square and fill in cells by advancing in a knight fashion to the right until blocked by a previous number.
  2. Move one cell right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
1
4
2
5
3
2
1 7
10 4
28
5 6
3 9
3
1 7 13
1011 4
28 14
12 5 6
3 9 15
4
19 1 7 13
101117 4
28 14 20
1218 5 6
3 9 15 16
5 KL5* 8 [(2D,1L),R]
19 25 1 7 13
101117 23 4
2128 14 20
121824 5 6
3 9 15 16 22

The Other Four 5x5 Loubère Knight-step Squares

A KL5* 23 [(2D,1L),R]
9 15 16 22 3
2517 13 19
111723 4 10
2814 20 21
18 24 5 6 12
 
B KL5* 13 [(2D,1L),R]
24 5 6 12 18
151622 3 9
1713 19 25
17234 10 11
8 14 20 21 2
 
C KL5* 3 [(2D,1L),R]
14 20 21 2 8
5612 18 24
16223 9 15
71319 25 1
23 4 10 11 17
 
D KL5* 18 [(2D,1L),R]
4 10 11 17 23
20212 8 14
61218 24 5
2239 15 16
13 19 25 1 7

A 7x7 Loubère Knight-step Square

The construction of a 7x7 Loubère Knight-step square KL7* 39 [(2D,1L),1R] from the broken yellow diagonal is shown below using the knight-step approach.

1
1 9
13 5
2 10
15 6 14
3 11
7 8
4 12
2
1 9 17 25
132122 5
2 10 18 26
1523 6 14
3 11 19 27
24 7 8 16
4 12 20 28 29
3
33 41 49 1 9 17 25
132122 30 38 46 5
42432 10 18 26 34
152331 39 47 6 14
44 3 11 19 27 35 36
24 32 40 48 7 8 16
4 12 20 28 29 37 45

Three 9x9 Loubère Knight-step Squares

Three 9x9 Loubère Knight-step square from the broken yellow diagonal are shown along with the accompanying table of 9 squares showing the triad sums of the left diagonal. Square A shows a typical knight and Right move.

A KL9* 23 [(2D,1L),1R]
516171 81111 213141
162636 374757 67776
627273 21222 324252
272838 485868 78717
64743 132333 435363
293949 596979 81819
75414 243444 545565
405060 70809 102030
51525 354546 566676
 
B KL9* 68 [(2D,1L),1R]
152535 454656 66765
617181 11121 314151
263637 475767 77616
72732 122232 425262
283848 586878 71727
74313 233343 536364
394959 69798 181929
41424 344454 556575
506070 80910 203040
 
C KL9* 32 [(2D,1L),1R]
607080 91020 304050
253545 465666 76515
71811 112131 415161
363747 576777 61626
73212 223242 526272
384858 68787 172728
31323 334353 636474
495969 79818 192939
142434 445455 65754
9x9 Cell Values and S of K[(2D,1L),1R]
center ValueS + dd
2345081
683690
32288-81
7745081
413690
5288-81
5045081
143690
59288-81

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 KL7* 39 [(2D,1L),1R] one can move up the right diagonal on a plane of four KL7* 39 [(2D,1L),1R] 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 Loubère Knight-step square method (Part II). To return to homepage.


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