L - Wings: examples and exemplar {2 digit - 3 strong links}

Advanced methods and approaches for solving Sudoku puzzles

L - Wings: examples and exemplar {2 digit - 3 strong links}

Postby StrmCkr » Sat Dec 27, 2014 12:38 am

This is not my own technique it has been on a few sites for a while, however the information for it was limited.

I built this page in hopes to standardize the L-Wing technique.

an L wing is constructed using grouped or none grouped candidates from 2 digits containing a minimal of three strong-links.
{note: a 4th strong-link may be noted at a weak-link interface, this occurs from arrangement of the box's,row,cols used}

a generalized L-wing is possible expansion removes the three strong-link restraint through the weak-link via additional strong-links on one candidate.
if these are used the designation gL- Wing is warranted.

information Ive located originally identified L2 -wings and L-3 wings; however L3 wings directly overlap another solving technique {hybrid wings} and use 3 candidates Ive excluded these types;
Code: Select all
L3-Wing:  (X)a = (X   -  Y)b = (Y-Z)c = (Z)d     "a" and "d" in same unit; a<>Z, d<>X
H1-Wing:  (X)a = (X   -  Y)b = (Y-Z)c = (Z)d     "a" and "d" in same unit; a<>Z, d<>X


Which leaves only 1 scenario: the L2-wing which can be read in two directions, and because this is the only variation remaining the designation "2" can be dropped.

L-Wing: (X)a = (X)b - (X)c = (X-Y)d = (Y)e =>> "a" and "e" in same unit; a<>Y, e<>X
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ { or } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
L -Wing: (Y)e = (Y -X)d = (X)c -(x)b = (x)a =>> "a" and "e" in same unit; a<>Y, e<>X


L-Wings are classed by two types :

Type 1: none grouped candidates {all strong links} so that any Box,Row,and Col sector can be used for any of the three strong links: for simplicity I'll only list an exemplar instead of generating all possible base type configurations.

Type 2: grouped candidates where by weakly linked candidates become strong links, so that a limited selection of combinations can be used for any of the three strong links, for simplicity I'll only list a few exemplars instead of generating all base type configurations.

Code: Select all
Type 1: None Grouped Candidates: (X)Col 2,8, (Y) Box 9 ->>
--------------------------------------
|  .  /    . |  .  .  . |  .    /  . |
|  .  x    . |  .  .  . |  .    x  . |
|  .  /    . |  .  .  . |  .    /  . |
--------------------------------------
|  .  /    . |  .  .  . |  .    /  . |
|  .  /    . |  .  .  . |  .    /  . |
|  .  /    . |  .  .  . |  .    /  . |
--------------------------------------
|  .  /    . |  .  .  . |  /  xy+  / |
|  .  x-y  . |  .  .  . |  /   / y-x |
|  .  /    . |  .  .  . |  /   /   / |
--------------------------------------

Code: Select all
 Type 2: Grouped candidates  (X)Col 2,3 (Y) Col 3         Type 2a: Grouped Candidates  (X) Box 6,Row 1 (Y) Col 4               
--------------------------------------                           ----------------------------------                         
|  .  x    x |  .  .  . |  .    .  . |                           |  /  /  / | xy+ /  / |  x  /  / |                           
|  .  x    x |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  .  .  . |                       
|  .  x    x |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  .  .  . |                         
--------------------------------------                           ----------------------------------                         
|  .  /    / |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  x  /  / |                       
|  .  /  xy+ |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  x  /  / |                       
|  .  /    / |  .  .  . |  .    .  . |                           |  .  .  . | y-x .  . |  x x-y / |                         
--------------------------------------                           ----------------------------------                         
|  .  /    y |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  .  .  . |                         
|  .  x-y  y |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  .  .  . |                         
|  .  /    y |  .  .  . |  .    .  . |                           |  .  .  . |  /  .  . |  .  .  . |                         
--------------------------------------                           ----------------------------------

Code: Select all
Type 2b: Grouped candidates  (X) Row2,8 (Y)Box 1
----------------------------------
|  .  .  . |  /  y  / |  .  .  . | 
|  /  /  / |  /  y xy |  /  x  / |
|  .  .  . |  /  y  / |  .  .  . |
----------------------------------
|  .  .  . |  .  .  . |  .  .  . | 
|  .  .  . |  .  .  . |  .  .  . |
|  .  .  . |  .  .  . |  .  .  . |
----------------------------------
|  .  .  . |  .  .  . |  .  .  . | 
|  /  /  / |  / x-y / |  /  x  / |
|  .  .  . |  .  .  . |  .  .  . |
----------------------------------     

    'XY+' means the cell must contain both 'X' and 'Y' candidates, and possibly others
    '-X' means 'X' is eliminated
    '-Y' means 'Y' is eliminated
    '/' indicates cell is devoid of candidate 'X' or 'Y' depending on which candidate is using that sector, for clarity Candidate Sectors is included at the top of the exemplar.

Thanks to daj95376, for information examples and exemplars regarding this technique.
Last edited by StrmCkr on Fri Jan 09, 2015 7:48 am, edited 4 times in total.
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Re: L - Wings: examples and exemplar {2 digit - 3 strong lin

Postby StrmCkr » Sat Dec 27, 2014 9:18 pm

reserved for minimal examples:

L wing found after basic techniques are applied and its application reduces the puzzle to singles only:

Code: Select all
9...5..3.4..1..9.8..6.9.4.2...97..2...42168...9..3....8.3...2..7....9.8..49.8...6

{Type 1} L-Wing: (4)r6c4 = (4-6)r1c4 = r1c7 - r2c8 = (6)r6c8 => r6c8<>4
{Type 1} L-Wing: (4)r6c4 = (4-6)r1c4 = r2c5 - r2c8 = (6)r6c8 => r6c8<>4
{Type 2a} L-Wing: (4)r6c4 = (4-6)r1c4 = r1c7 - r46c7 = (6)r6c8 => r6c8<>4


{ if you happen to have examples of this technique feel free to pm me the grid's and ill add it to this list}

Non - minimal examples
Code: Select all
...3.2.7.7..9....5..6...4...948.....2.1...5.7.....198...8...7..4....5..3.6.2.9...

{Type 1} L-Wing (5)r6c4 = (5-7)r3c4 = r8c4 - r8c2 = (7)r6c2 => r6c2<>5
{ Type 2b} L-Wing (6)r12c5 = (6-8)r2c6 = r2c7 - r8c7 = (8)r8c5 => r8c5<>6
Last edited by StrmCkr on Tue Dec 30, 2014 8:06 am, edited 4 times in total.
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Re: L - Wings: examples and exemplar {2 digit - 3 strong lin

Postby denis_berthier » Mon Dec 29, 2014 6:02 am

Hi StrmCkr

Your L-Wing example patterns can be seen as 3D bivalue-chains[3] or whips[3]:

{Type 1} L-Wing: (4)r6c4 = (4-6)r1c4 = r1c7 - r2c8 = (6)r6c8 => r6c8<>4
is a bivalue-chain[3]: c4n4{r6 r1} - r1n6{c4 c7} - c8n6{r2 r6} ==> r6c8 ≠ 4

{Type 1} L-Wing: (4)r6c4 = (4-6)r1c4 = r2c5 - r2c8 = (6)r6c8 => r6c8<>4
is a bivalue-chain[3]: c4n4{r6 r1} - b2n6{r1c4 r2c5} - c8n6{r2 r6} ==> r6c8 ≠ 4


{Type 2a} L-Wing: (4)r6c4 = (4-6)r1c4 = r1c7 - r46c7 = (6)r6c8 => r6c8<>4
is a whip[3]: c4n4{r6 r1} - r1n6{c4 c7} - b6n6{r4c7 r6c8} ==> r6c8 ≠ 4, where n6r4c7 is a t-candidate in the 3rd cell
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