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

A Loubere square

A Discussion of the New Methods

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

This method differs from the previous due to the two ways the knight may move. It may move down 2, right 1, i.e., (2D,1R) or up 2, right 1, i.e., (2U,1R).

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For every square not divisible by 3, the (2U,1R) set forms magic squares for the left diagonals, while the (2D,1R) forms semi-magic squares in the left diagonal.
  3. For Loubère Knight-step where n is divisible by 3, i.e., 3(2n + 1), the sum of the left diagonal is always magic for the (2D,1R) squares and semi-magic for the (2U,1R) squares.

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

  1. To generate these square, place a 1 into any of the cells on the yellow diagonal of a 5x5 square and fill in cells by advancing in a knight (2 cells up,1 cell right) 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 the group.

The 5x5 Loubère Knight-step Magic Squares

A KL5* 13 [(2U,1R),1D]
10 18 1 14 22
11247 20 3
17513 21 9
23619 2 15
4 12 25 8 16
 
B KL5* 20 [(2U,1R),1D]
12 25 8 16 4
18114 22 10
24720 3 11
51321 9 17
6 19 2 15 23
 
C KL5* 22 [(2U,1R),1D]
19 2 15 23 6
25816 4 12
11422 10 18
7203 11 24
13 21 9 17 5
 
D KL5* 4 [(2U,1R),1D]
21 9 17 5 13
21523 6 19
8164 12 25
142210 18 1
20 3 11 24 7
 
E KL5* 6 [(2U,1R),1D]
3 11 24 7 20
9175 13 21
15236 19 2
16412 25 8
22 10 18 1 14

The Set of 5x5 Knight-step Loubère Semi-magic Squares [(2D,1R),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 (2 cells down,1 cell right) 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 the group having left diagonal sums of 65, 55, 70, 60 and 75, respectively, where the last four sums are semi-magic.

The 5x5 Loubère Knight-step Semi-magic Squares

F KL5* 13 [(2D,1R),1D]
23 12 1 20 9
4187 21 15
102413 2 16
11519 8 22
17 6 25 14 3
 
G KL5* 21 [(2D,1R),1D]
6 25 14 3 17
12120 9 23
18721 15 4
24132 16 10
5 19 8 22 11
 
H KL5* 9 [(2D,1R),1D]
19 8 22 11 5
25143 17 6
1209 23 12
72115 4 18
13 2 16 10 24
 
I KL5* 17 [(2D,1R),1D]
2 16 10 24 13
82211 5 19
14317 6 25
20923 12 1
21 15 4 18 7
 
J KL5* 5 [(2D,1R),1D]
15 4 18 7 21
161024 13 2
22115 19 8
3176 25 14
9 23 12 1 20

Two 7x7 Loubère Knight-step Squares

Two 7x7 Loubère Knight-step square from the broken yellow diagonal are shown. Square K belongs to the set of (2U,1R) and L belongs to (2D,1R) set. Typical knight and break moves are shown.

K KL7* 25 [(2U,1R),1D]
132340 11835 45
213148 92636 4
22397 173444 12
30478 25423 20
38616 334311 28
461424 41219 29
51532 491027 37
 
L KL7* 25 [(2D,1R),1D]
463116 14227 12
53924 94335 20
134732 17236 28
21640 251044 29
221448 33183 37
30157 412611 45
38238 493419 4
7x7 Cell Values and S
center Value (2U,1R),1DS + ddcenter Value (2D,1R),1DS + dd
251750251750
3417502161-14
3617503519621
451750121827
5175038168-7
14175015154-21
1617504818914

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. The first square is composed of triad sums for the left diagonal, while the second square is magic for each of the left diagonals. Squares K and L show a typical knight and up or down moves. Square M belongs to the set of (2U,1R) and N belongs to (2D,1R) set. Typical knight and break moves are also shown.

M KL9* 41 [(2U,1R),1D]
162849 70122 435576
263859 801132 53655
364869 92142 637515
375879 103152 64425
47688 204162 741435
577818 305172 32445
67719 406173 133446
771729 50712 234456
62739 608112 335466
 
N KL9* 41 [(2D,1R),1D]
775839 20172 533415
66849 301173 634425
167859 40212 645435
26769 503112 745545
361779 604122 36546
37278 705132 137556
472818 806142 23466
573819 97152 331476
674829 108162 43245
9x9 Cell Values and S
center Value (2U,1R),1DS + ddcenter Value (2D,1R),1DS + dd
413690413690
5245081123690
63288-81643690
653690443690
7645081153690
6288-81673690
173690383690
1945081183690
30288-81703690

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* 25 [(2U,1D),1D] one can move up the right diagonal on a plane of four KL7* 39 [(2U,1D),1D] 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 III). To visit the new Knight version of Méziriac squares (Part I) or To return to homepage.

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