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

A Loubere square

A Discussion of the New Methods

This is a continuation of Part I which details the set of Loubère Knight-step squares. The new Loubère type squares of Part II, KLn* (center cell#) [Knight move, 1U or 1R] where KLn* signifies a Knight-step nxn Loubère type square with the center cell number of the square, with a knight move of (2D,1L) or (2L,1D), followed by breaking either up or down. Note that these apply to those squares generated with 1 on the yellow diagonal see Part I.

  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 are produced where the sum of the left diagonal S equals S + d, and d is equal to either -n2, 0 or n2.

The Set of 5x5 Knight-step Loubère Magic Square [(2D,1L),1U]

  1. To generate these square, place a 1 into any of the cells on the yellow diagonal row of a 5x5 square and fill in cells by advancing in a knight fashion down until blocked by a previous number. One square shows a typical knight and break move.
  2. Move one cell up.
  3. Repeat the process until the square is filled.
  4. Generate the five squares belonging to a group.

The 5x5 Loubère Knight-step Squares

A KL5* 19 [(2D,1L),1U]
22 14 1 18 10
16825 12 4
15219 6 23
92113 5 17
3 20 7 24 11
 
B KL5* 12 [(2D,1L),1U]
20 7 24 11 3
14118 10 22
82512 4 16
2196 23 15
21 13 5 17 9
 
C KL5* 10 [(2D,1L),1U]
13 5 17 9 21
72411 3 20
11810 22 14
25124 16 8
19 6 23 15 2
 
D KL5* 3 [(2D,1L),1U]
6 23 15 2 19
5179 21 13
24113 20 7
181022 14 1
12 4 16 8 25
 
E KL5* 21 [(2D,1L),1U]
4 16 8 25 12
23152 19 6
17921 13 5
11320 7 24
10 22 14 1 18

The Set of 5x5 Knight-step Loubère Magic Square [(2D,1L),1D]

  1. To generate these square, place a 1 into any of the cells on the yellow diagonal row of a 5x5 square and fill in cells by advancing in a knight fashion down until blocked by a previous number. One square shows a typical knight and break move.
  2. Move one cell down.
  3. Repeat the process until the square is filled.
  4. Generate the five squares belonging to a group.

The 5x5 Loubère Knight-step Squares

F KL5* 13 [(2D,1L),1D]
9 20 1 12 23
15217 18 4
16213 24 10
22819 5 11
3 14 25 6 17
 
G KL5* 18 [(2D,1L),1D]
14 25 6 17 3
20112 23 9
21718 4 15
21324 10 16
8 19 5 11 22
 
H KL5* 23 [(2D,1L),1D]
19 5 11 22 8
25617 3 14
11223 9 20
7184 15 21
13 24 10 16 2
 
I KL5* 3 [(2D,1L),1D]
24 10 16 2 13
51122 8 19
6173 14 25
12239 20 1
18 4 15 21 7
 
J KL5* 8 [(2D,1L),1D]
4 15 21 7 18
10162 13 24
11228 19 5
17314 25 6
23 9 20 1 12

Two 9x9 Loubère Knight-step Squares

Two 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. Squares K and L show a typical knight and up or down moves.

K KL9* 51 [(2D,1L),1U]
765543 22170 492816
665433 128160 39276
564423 27150 291777
463413 736140 19767
45243 725130 187857
351474 624120 86847
25464 523110 795837
157563 42219 694836
56553 321180 593826
 
L KL9* 41 [(2D,1L),1D]
153453 72120 395877
254463 731130 49686
355464 22140 597816
455574 123150 69726
46653 224160 791736
567513 325170 82737
66423 426180 182847
761433 52719 193857
52443 628110 294867
9x9 Cell Values of two types of Knight Squares and their left diagonal Sums
center Value (2D,1L),1US + ddcenter Value (2D,1L),1DS + dd
5145081413690
4036905045081
29288-8159288-81
2745081683690
1636907745081
5288-815288-81
7545081143690
6436905345081
62288-8132288-81

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 LK7* 39 [(2D,1L),1R] one can move up the right diagonal on a plane of four LK7* 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 knight-step squares (Part II). The next section deals with a new Loubère Knight-step method (Part III). To return to homepage.


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