The New Sudoku Players' Forum

by **StrmCkr** » Sat Dec 24, 2016 8:19 am

how the barn was built:https://sysudoku.com/2016/02/16/barn-built/

since i had multi conversations with the author seen in this blog, Ive decided I'll accept his nomination for the named technique and develop it further

a barn is built on the idea that a subset can be "bent" to fit out side of single box/Row/Col restrictions but instead resided inside shared sectors via shared digits so that the shared digits "lock" some of the candidates into specific cells in a way so that peers of these cells can not hold said candidates.

this thread hopes to amalgamate several low level techniques with some higher lvl functions to correlate eliminations seen but often unaccounted without resorting to a higher function logic to cover the deduction.

From what i can tell, this list is potentially but not limited to:

I discovered this concept but lacked the terms to express the deduction and concept easily way back in my wxyz-thread

when searching for a set of N candidates found In N Cells
build a relation ship between the N cells by aligning them to sectors
and forming a restricting link reducing the placement arrangement of N-1 candidates into n-1 cell.
all peer cells of a non restricted common candidates that sees each candidate in N cells can be excluded.
if all candidates are restricted to N cells then all peer cells that see each of said candidate may be excluded.

working Concept:
N Cells holding N candidates:
Search for a group of N Cells that are all peers of a starting point each holding up to N candidates
search for sector A & Sector B
where A & B contain all N cells
where A <> B, and B <> A
A <> empty and B <> empty

next check Linking sectors:
sector(s) "L "
where L contains "R" candidates limited to each L sector

type 1: R = N-1 candidates
Z = A Candidate not found in R
both sectors A&B contain cells for digit "z"
elimination: { almost locked}
all peer cells that see each "Z" candidate in sectors A&B may be excluded.

Type 2 elimination: dual linked /sue de coq ie subset counting
R = N candidates
all cells that are peer of each "R" candidate, may be excluded.

Type 3: elimination R= N -2 {almost almost locked }
theorized
Z = missing digits from R
all peer cells for "z" that also see a situation that can place multiple or restrict multiple digit locations with in the subset may be eliminated.

example:

Code: Select all: Wxyz - Wing {extended} . 124 . | . . . | 13 . . . . . | . . . | . . . 24 -24 -24| . . . | . 234 . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

Type 4: theorized
N cells holding (N-M) Candidates where all N cells are missing the same Candidate(s)
elimination :

Code: Select all: ---------------------------------- abc -b -b | bc . . | -b -b -b . . . | . . . | . . . . . ab | . bc . | . . . ----------------------------------

Eventual further work and Goal:
i would like to make this project also apply to more then just direct peer cells from a starting point as this makes barns strictly limited to 1-9 cell "wings" that in habit box+Row/col configurations, and to cover some of the examples listed below.

examples:
example: XY - Wing

Code: Select all: . 12 . | . . . | . 23 . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . 13 . | . . . | . -3 . . . . | . . . | . . . search for 3 Cells Candidate set [123] Cells [ R1C2,R1C8,R8C2] type 1 examples: 1{ A=C2 B=C8 L=R1 R=2 Z=3} 2{A=R1 B=R7 L=C1 R=1 Z=3} eliminations 3 @ R8C8

Code: Select all: wxyz-wing . wz . | . . . | . . . -Z wxyz -Z | . xyz . | . xyz . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . search for 4 Cells with candidate set size 4 Cells [1,10,13,16] A:=0 B:=1 R:= 1 L:= 10 Z:= "Z" <> @ Cells 9,11

Code: Select all: . 124 . | . . . | . 13 . . . . | . . . | . . . . 24 . | . . . | . . -------------------------------- . . . | . . . | . . . . -2 . | . . . | . 23 . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . search for 4 Cells with candidate set size 4 Canddiate set [1234]

Code: Select all: Wxyz-Wing -- basic elimination shown . . . | . . . | . . . -3 678 83 | . . . | . 78 . . 36 . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . additionally the following are applicable Wxyz - double linked rule: A=r2c238 {3678}, B=r3c2 {36}, X=3,6 => r1c123,r23c1,r3c3,r456789c2<>6, r1c123,r23c1,r3c3<>3, r2c145679<>7, r2c145679<>8

Edit: updated contextual wording of the project goals and present situation.
Jan 5th 2017

by **rjamil** » Mon Dec 26, 2016 2:11 pm

Hi StrmCkr,

Allow me to understand BarnS with few (stupid) questions/suggestions as follows:

1. Is it necessary to include 3 cells/digits for BarnS?
Suggest to start BarnS concept from >= 4 cells/digits combinations as XY- & XYZ-Wings do not have more types/ways to consider (as 2 c/d already not considered).

StrmCkr wrote:
Code: Select all
. 124 . | . . . | . 13 . . . . | . . . | . . . . 24 . | . . . | . . . -------------------------------- . . . | . . . | . . . . -2 . | . . . | . 23 . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . search for 4 Cells with candidate set size 4 Canddiate set [1234] Cells[R1C2,R1C8,R3C2,R5C8] elimination = 2 @ R5C2

2. Is R3C2 [24] can be place anywhere in C2 free cell, i.e., B147C2?

R. Jamil

by **Sudtyro2** » Mon Dec 26, 2016 10:46 pm

Hi StrmCkr,
I'm interested in your BarnS technique, but haven't yet studied the algorithms for the two types listed. In the meantime, I'm curious to know if you have any example subsets of the N-digit/N-cell variety that cannot be solved with the Subset-Counting technique. Here, I use the term 'digit' as a 'candidate.'

I ask this question because, as I understand the older method, one need only visually examine the grid for any digit external to the subset that can 'see' all occurrences of the digit within the subset. If that happens to a multiplicity-2 digit, as it does in your samples, it would reduce the total 'multiplicity' of the subset to a value below the cell count. That would then be a contradiction and would require the external digit to be removed.

The term, multiplicity, refers to the maximum number of placements of a particular digit within a subset. Total multiplicity is the sum of all multiplicities within the subset. The digit of interest in each example above has a multiplicity of two. All others have a multiplicity of one.

SteveC

by **StrmCkr** » Tue Dec 27, 2016 6:43 pm

thanks for the interest Steve, i highly doubt this will bring anything new to the table elimination wise:
its more of a different approach for "bending" subsets around R/C/B but not limited to the B + R/C restraints as seen in "wings"

if you already have:

which also nicely covers
subset counting

but, cannot cover the next cases until you start using als-chain rules
distributed-disjoint-subsets

the closest contextual idea is the DDS to what i'm trying to accomplish
on inspection DDs is different as N cells = # N digit links

I'm curious to know if you have any example subsets of the N-digit/N-cell variety that cannot be solved with the Subset-Counting technique.

I do not have subset or DDS logic coded ATM so I cannot compare these techniques conclusively:

in essence i am trying to come up code that explains the "wing" effect of the following with out using multiple different techniques

Code: Select all: . 124 . | . . . | 13 . . . . . | . . . | . . . 24 -24 -24| . . . | . 234 .

Fits by cell count & digit count definition for a wxyz but didn't fit into the formation clause of B+R/C or R/C + C/R of a typical wing.

the sectors A&B used have no als-xz elimination
and the elimination is only derived via Als-xy rules using a third sector

it doesn't fit into DDS by description as 1 is locked to R1, 3 is locked to b3
and 2,4 are not covered
it would fall into Almost Almost DDS strategy found in the same link above. {where unlocked digits eliminate from all shared peers )

side note:
BARN idea currently coded include these cases but currently performs no eliminations :

as a search for a set of 4 digits on 4 cells can find subset's where M candidate is missing from all found cells.

Code: Select all: +----------+-----------+----------+ | abc . . | bc . . | -c -c -c | | . . . | . . . | . . . | | . . ac | . bc . | . . . | +----------+-----------+----------+

inversely: N candidates in N+{M} cells

by **rjamil** » Tue Jan 03, 2017 1:34 pm

Hi StrmCkr,

I am working on your defined pattern strategy named Barn.

I have studied WXYZ-wings (i.e., 4-cell with exactly 4 values only) like strategy that covers within Barn. I have concluded two main types, i.e., one with pivot cell contains all four values and other with pivot cell contains only three values along with three pincer cells contain exactly four values as follows:

Code: Select all: --------------+-------------+----------- 01) . wz . | . . . | . . . -Z wxyz -Z | . xz yz | . . . . . . | . . . | . . . --------------+-------------+----------- 02) . xz yz | . . . | . . . -Z wxyz -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 03) . wz . | . . . | . . . -Z wxyz -Z | . xz . | . . yz . . . | . . . | . . . --------------+-------------+----------- 04) . xz . | . . . | . . . -Z wxyz -Z | . wz . | . . . . yz . | . . . | . . . --------------+-------------+----------- 05) . wz . | . . . | . . . -Z wxyz -Z | . xy yz | . . . . . . | . . . | . . . --------------+-------------+----------- 06) . xy yz | . . . | . . . -Z wxyz -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 07) . wz . | . . . | . . . -Z wxyz -Z | . xy . | . . yz . . . | . . . | . . . --------------+-------------+----------- 08) . xy . | . . . | . . . -Z wxyz -Z | . wz . | . . . . yz . | . . . | . . . --------------+-------------+----------- 09) . wz . | . . . | . . . -Z wxyz -Z | . xy xyz | . . . . . . | . . . | . . . --------------+-------------+----------- 10) . xy xyz | . . . | . . . -Z wxyz -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 11) . wz . | . . . | . . . -Z wxyz -Z | . xy . | . . xyz . . . | . . . | . . . --------------+-------------+----------- 12) . xy . | . . . | . . . -Z wxyz -Z | . wz . | . . . . xyz . | . . . | . . . --------------+-------------+----------- 13) . wz . | . . . | . . . -Z wxyz -Z | . xz xyz | . . . . . . | . . . | . . . --------------+-------------+----------- 14) . xz xyz | . . . | . . . -Z wxyz -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 15) . wz . | . . . | . . . -Z wxyz -Z | . xz . | . . xyz . . . | . . . | . . . --------------+-------------+----------- 16) . xz . | . . . | . . . -Z wxyz -Z | . wz . | . . . . xyz . | . . . | . . . --------------+-------------+----------- 17) . wz . | . . . | . . . -Z wxyz -Z | . xyz xyz | . . . . . . | . . . | . . . --------------+-------------+----------- 18) . xyz xyz | . . . | . . . -Z wxyz -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 19) . wz . | . . . | . . . -Z wxyz -Z | . xyz . | . xyz . . . . | . . . | . . . --------------+-------------+----------- 20) . xyz . | . . . | . . . -Z wxyz -Z | . wz . | . . . . xyz . | . . . | . . . --------------+-------------+-----------

1) WXYZ-Wing strategy is based on four unsolved cells (abcd) align within a chute that contain exactly four values (wxyz);
2) One pivot cell (a) contains all four values (wxyz) and three pincer cells (bcd) // contain exactly four values - need not to be check at this moment;
3) First pincer cell (b) align with pivot cell's (a) box (or line) and other two pincers cell (cd) align with pivot cell's (a) line (or box) but not first pincer cell's (b) line;
4) First pincer cell (b) contains only two values (wz) and other two pincers cell (c | d) contain exactly three values (xyz);
5) First pincer cell and other two pincers cell (b & (c | d)) contain exactly one common value (z);
6) First pincer cell and other two pincers cell (b & (c | d)) common value (z) may be eliminated from cells align with pivot cell (a) box and other two pincers cell (cd) (or first pincer cell (b)) common line.

Code: Select all: --------------+-------------+----------- 21) . wz . | -Z -Z -Z | . . . -Z wxy -Z | . xz yz | . . . . . . | . . . | . . . --------------+-------------+----------- 22) . xz yz | -Z -Z -Z | . . . -Z wxy -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 23) . wz . | . . . | . . . -Z wxy -Z | . xz . | . . yz . . . | . . . | . . . --------------+-------------+----------- 24) . xz . | . . . | . . . -Z wxy -Z | . wz . | . . . . yz . | . . . | . . . --------------+-------------+----------- 25) . wz . | -Z -Z -Z | . . . -Z wxy -Z | . xy yz | . . . . . . | . . . | . . . --------------+-------------+----------- 26) . xy yz | -Z -Z -Z | . . . -Z wxy -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 27) . wz . | . . . | -Z -Z -Z -Z wxy -Z | . xy . | . yz . . . . | . . . | . . . --------------+-------------+----------- 28) . xy . | . . . | . . . -Z wxy -Z | . wz . | . . . . yz . | -Z -Z -Z | . . . --------------+-------------+----------- 29) . wz . | -Z -Z -Z | . . . -Z wxy -Z | xy xyz . | . . . . . . | . . . | . . . --------------+-------------+----------- 30) . xy xyz | -Z -Z -Z | . . . -Z wxy -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 31) . wz . | . . . | -Z -Z -Z -Z wxy -Z | . xy . | . xyz . . . . | . . . | . . . --------------+-------------+----------- 32) . xy . | . . . | . . . -Z wxy -Z | . wz . | . . . . xyz . | -Z -Z -Z | . . . --------------+-------------+----------- 33) . wz . | -Z -Z -Z | . . . -Z wxy -Z | xz xyz . | . . . . . . | . . . | . . . --------------+-------------+----------- 34) . xz xyz | -Z -Z -Z | . . . -Z wxy -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 35) . wz . | . . . | . . . -Z wxy -Z | . xz . | . xyz . . . . | . . . | . . . --------------+-------------+----------- 36) . xz . | . . . | . . . -Z wxy -Z | . wz . | . . . . xyz . | . . . | . . . --------------+-------------+----------- 37) . wz . | -Z -Z -Z | . . . -Z wxy -Z | xyz xyz . | . . . . . . | . . . | . . . --------------+-------------+----------- 38) . xyz xyz | -Z -Z -Z | . . . -Z wxy -Z | . wz . | . . . . . . | . . . | . . . --------------+-------------+----------- 39) . wz . | . . . | . . . -Z wxy -Z | . xyz . | . xyz . . . . | . . . | . . . --------------+-------------+----------- 40) . xyz . | . . . | . . . -Z wxy -Z | . wz . | . . . . xyz . | . . . | . . . --------------+-------------+-----------

1) WXYZ-Wing strategy is based on four unsolved cells (abcd) align within a chute that contain exactly four values (wxyz);
2) One pivot cell (a) contain only three values (wxy) and three pincers cell (bcd) // contain exactly four values (wxyz) - may not be check at this moment;
3) First pincer cell (b) align with pivot cell's (a) box (or line) and other two pincers cell (cd) align with pivot cell's (a) line (or box) but not first pincer cell's (b) line;
4) First pincer cell (b) contains only two values (wz) and other two pincers cell (c | d) contain exactly three values (xyz);
5) First pincer cell and other two pincers cell (b & (c | d)) contain exactly one common value (z);
6) First pincer cell and other two pincers cell (b & (c | d)) common value (z) may be eliminated from cells that see those pincers cell that contain common value (z).

Note: these exemplars are not fully covered the Barn strategy but may be covered fixed patterns of WXYZ-Wings that will be easily identified/detected by either human being or computer program.

All members are welcome to share their ideas/proposals with StrmCkr here in order to discover and finalize all patterns combinations of Barn.

R. Jamil

Updated on 20170104:
- Corrected wordings and added notations for better understanding.
- Whereas "&" means intersection and "|" means union.
- Thanks StrmCkr for pm the acknowledgment of my work.

by **StrmCkr** » Thu Jan 05, 2017 12:45 pm

updated first post:
added working barns concept to solver page

by **eleven** » Thu Jan 05, 2017 7:34 pm

Many familiar patterns here, but a simple one i recently saw again seems not to be covered, cause it has 3 candidates in 4 cells:

Code: Select all: ab ab ----- ac abc

Eliminate a in the abc column. You can call it an almost xy-wing (ab-ac-cb+a), but that sounds complicated.

by **rjamil** » Thu Jan 05, 2017 8:36 pm

Hi eleven,

eleven wrote:Many familiar patterns here, but a simple one i recently saw again seems not to be covered, cause it has 3 candidates in 4 cells:
Code: Select all
ab ab ----- ac abc

Eliminate a in the abc column. You can call it an almost xy-wing (ab-ac-cb+a), but that sounds complicated.

If I am correct then this is XYZ-Wing Hybrid pattern, discovered by Pritt Galford here.

R. Jamil

by **eleven** » Fri Jan 06, 2017 9:42 pm

Yes, but this not only sounds complicated, it is by far the most complicated way i am aware to describe it.
[Edit:] Now i see, that i did not post it right. What i had in mind, was in 4 boxes, not in 2:

Code: Select all: ab . . | ab . . -------------- ac . . | abc . .

Since the pattern here is in the same columns, the deduction is the same, as AIC:
(ab=c)r34c4-(c=b)r34c1-(b=a)r3c4 => not a elsewhere in c4
[oops:]corrected
This is not covered by the XYZ-Wing Hybrid, as far as i can see.

by **rjamil** » Thu Feb 02, 2017 4:28 pm

Hi StrmCkr,

In continuation of my above mentioned post, I had stopped further studying due to the topic still not fully covered the matter. However, now I have decided to start coding for the types that I have already covered in my above mentioned post.

As I have already categorized only 40 exemplars based on pivot cell's number of digits and define three pincers as follows:

Pincers:-
- First pincer shares pivot box;
- Second pincer shares pivot line but not pivot box and first pincer line; and
- Third pincer shares pivot either box but not second pincer line; or pivot line but not pivot box.

WXYZ-Wings Type 1:- (20 exemplars)
Pivot cell contains all four digits and number of elimination cells are two.

WXYZ-Wings Type 2a:- (14 exemplars)
Pivot cell contains three digits and number of elimination cells are five (instead of two).

WXYZ-Wings Type 2b:- (6 exemplars)
Pivot cell contains three digits and number of elimination cells are two.

At the moment I have completed the logic, coded as entirely separate module accordingly and under rigorous testing various scenarios. Will share the source code once satisfied the performance.

R. Jamil

by **StrmCkr** » Fri Feb 03, 2017 2:09 am

That's alright the idea I have coded and shared covers,
From pivot point connecting to 2-8 "wings" cells with in a box+row or box +col intersection.

For n size set of digit with in the n cells, that code works pretty good atm.

Still haven't figured out how to get it to expand into row +row or row+col, col+col types with out massive code errors.

Atm I'm working on my Als xz code as it also features the same hiccup as this code
Ie overlapping sets creates errors if I have them activated once I solve that part I'll continue the barn thread.
And attempt to get it to cover the project concepts.

by **rjamil** » Sat Feb 04, 2017 3:23 pm

Hi StrmCkr,

I have covered WXYZ-Wings that exist within box-line (or chute) wise.

Now I have just enumerated WXYZ-Wings that exist within row-column or column-row wise and only one elimination cell.

Got just four variations that satisfied the conditions:

Code: Select all: ----------+----------+---------- 1) . . . | . . . | . . . . wxy . | xy yz . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . wz . | . -z . | . . . . . . | . . . | . . . ----------+----------+---------- 2) . . . | . . . | . . . . wxy . | xy . . | . yz . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . wz . | . . . | . -z . . . . | . . . | . . . ----------+----------+---------- 3) . . . | . . . | . . . . wxy . | xy xyz . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . wz . | . -z . | . . . . . . | . . . | . . . ----------+----------+---------- 4) . . . | . . . | . . . . wxy . | xy . . | . xyz . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . wz . | . . . | . -z . . . | . . . | . . . ----------+----------+----------

Maybe there are more scenarios that I missed and will also satisfied the 4-cells and 4-digits within row-column and column-row wise.

Also, perhaps the above 4-cells and 4-digits scenario derives following steps that will cover for N-cells and N-digits scope (where N = 3 to 9):

- Only one pincer cell within pivot row contains z value;
- Only one pincer cell witihn pivot column contains z value;
- z may be eliminated from cell that see row and column pincers containing z value.

R. Jamil

by **StrmCkr** » Sun Feb 05, 2017 7:36 pm

here you go, a row + row example that's not based on a pivot seeing all cells. {which is where id like this eventually to go}

Code: Select all: . . . | . . . | . . . . 14 . | . . . | -4 -4 -4 -4 -4 -4 | . . . | . 34 . -------------------------------- . 12 . | . . . | . 23 . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

yes the examples you have shown are complete for the pivot seeing every cell.

for me these are covered for size N cells all seeing a pivot as coded in my barns program.
the example posted herein is the ones i want my program to eventually find.

my current hangup is found with this example

Code: Select all: . . . | . . . | . . . 89 13 -9 | . . . | . 189 . . . 39 | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

i can find the -9 elimination above
Almost Locked Set XZ-Rule: A=r2c17 {189}, B=r2c2,r3c3 {139}, X=1, Z=9 => r2c3<>9

but it misses the following

Code: Select all: . . . | . . . | . . . 89 13 . | -8 -8 -8 | -8 189 -8 . . 39 | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . -------------------------------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

which happens when using overlapping sets
Almost Locked Set XZ-Rule: A=r2c17 {189}, B=r2c12,r3c3 {1389}, X=1, Z=8,9 => r2c345689<>8, r2c3<>9

by **eleven** » Sun Feb 05, 2017 11:38 pm

fine, that's now an official wzyz-wing

http://forum.enjoysudoku.com/post254823.html#p254823

by **rjamil** » Mon Feb 06, 2017 3:26 am

Hi,

Many thanks for the confirmation. Now I will start coding for WXYZ-Wings Row-Column wise.

I am relabeling the WXYZ-Wings Types (with suggested renaming as well) as follows:

Type 1:- (04 exemplars) (suggest WXY-Wings Type 1)
WXYZ-Wings Row-Column wise, Pivot contain three values with one elimination cell;

Type 2a:- (14 exemplars) (suggest WXY-Wings Type 2a)
WXYZ-Wings Box-Line wise, Pivot contain three values with five elimination cells;

Type 2b:- (06 exemplars) (Suggest WXY-Wings Type 2b)
WXYZ-Wings Box-Line wise, Pivot contain three values with two elimination cells;

Type 3:- (20 exemplars) (WXYZ-Wings Type 3)
WXYZ-Wings Box-Line wise, Pivot contain four values with two elimination cells.

The above relabeling types and renaming is suggested due matches with XY-Wings and XYZ-Wings types.

R. Jamil

The New Sudoku Players' Forum

Bent Almost Restricted N(aked)-Set {Barn}

Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted Naked Set {BarnS}

Re: Bent Almost Restricted Naked Set {BarnS}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}

Re: Bent Almost Restricted N(aked)-Set {Barn}