Unique Rectangle Gallery 2

[KyouyamaKazusa0805]

Link: http://forum.enjoysudoku.com/type-3-unique-rectangles-hidden-subsets-t4088-15.html#p28313

Intro

I like the idea of programming up at least some of the options to see what they buy, but before I do I want to make sure I'm not unnecessarily restricting possible elimination. I've updated the list of new eliminations based on previous posts and see some significant differences in how things are presented as well as adding some options which I knew were missing (UR+3K and UR+4K). Although I don't use the term ALS (the descriptions seemed easier by describing the key characteristics rather than using ALS terminology) clearly that is what is being used. Obviously the definitions can apply with "a" and "b" switched or with "x", "Y", etc switched.

UR + 2

UR + 2kx

Two cells in a line, one with an extra candidate, "x", and one with at least one other extra different candidate, "Y", plus "(b)(a)x" common to "abx" which can contain “a” and which can also contain "b" if common to the "ab" which is in line with "abY" => "a" can be removed from "abY".

ab     ab
abx    abY  (b)(a)x

UR + 2kd

Two diagonal cells one with an extra candidate, "x", and one with at least one other extra different candidate, "Y", plus "(a)(b)x" common to "abx" which can contain “b” and which can also contain "a" if common to "abY" => "a" can be removed "abY".

ab     abY
abx    ab   (a)(b)x

UR + 2KX

Two cells in a line, with extra candidates "X" and "Y" plus extra cells, "(b)(a)U...", which can contain “a”, such that U is a locked set which includes "X", "abX" is seen by all cells of "(b)(a)U..." which contain elements of "X", and "(b)(a)U..." can contain "b" if the "ab" which is in line with "abY" is seen by all of the cells of "(b)(a)U..." which contain "b" => "a" can be removed from "abY".

ab     ab
abX    abY  (b)(a)U...

UR + 2KD

Two diagonal cells, with extra candidates "X" and "Y" plus extra cells, "(a)(b)U...", which can contain “b”, such that U is a locked set which includes "X", "abX" is seen by all cells of "(a)(b)U..." which contain elements of "X", and "(a)(b)U..." can contain "a" if "abY" is seen by all cells of "(a)(b)U..." which contain "a" => "a" can be removed from "abY".

ab     abY
abX    ab   (a)(b)U...

UR + 3

UR + 3kx

Three cells, two with one extra candidate, "abx" and "abz", and one with at least one other extra candidate, "abY", plus "(a)(z)x" common to "abx" and which can contain "a" if common to "abY" and/or "z" if common to "abz" and "(a)(x)z", plus "(a)(x)z" common to "abz" and which can contain "a" if common to "abY" and "x" if common to "abx" and "(a)(z)x" => "a" can be removed "abY".

ab      abz  (a)(x)z
abx     abY  (a)(z)x

UR + 3kd

Three cells, two with one extra candidate, "abx" and "abz", and one with at least one other extra candidate, "abY", plus "(a)(b)(z)x" common to "abx" and which can contain "a" if common to "abY", "b" if common to "ab" and/or "z" if common to "abz" and "(a)(b)(x)z", plus "(a)(b)(x)z" common to "abz" and which can contain "a" if common to "abY", "b" if common to "ab", and/or "x" if common to "abx" and "(a)(b)(z)x" => "a" can be removed "abY".

abz     ab   (a)(b)(x)z
abx     abY  (a)(b)(z)x

UR + 3KX

Three cells with extra candidates "X", "Y", and "Z", plus extra cells, "aU...", such that U is a locked set which includes "X", "abX" is seen by all cells of "aU..." which contain elements of "X", and "abY" is seen by all cells of "aU..." which contain "a", plus additional extra cells "aV...", such that V is a locked set which includes "Z", "abZ" is seen by all cells of "aV..." which contain elements of "Z", and "abY" is seen by all cells of "aV..." which contain "a" => "a" can be removed from "abY". Note that "U", and "V" need not be disjoint.

ab      abZ  aV...
abX     abY  aU...

UR + 3KD

Three cells with extra candidates "X", "Y", and "Z", plus extra cells, "(a)(b)U...", such that U is a locked set which includes "X", "abX" is seen by all cells of "(a)(b)U..." which contain elements of "X", and "(a)(b)U..." can contain "a" if "abY" is seen by all cells of "(a)(b)U..." which contain "a" and/or "(a)(b)U..." can contain "b" if "ab" is seen by all cells of "(a)(b)U..." which contain "b", plus additional extra cells "(a)(b)V...", such that V is a locked set which includes "Z", "abZ" is seen by all cells of "(a)(b)V..." which contain elements of "Z", and "(a)(b)V..." can contain "a" if "abY" is seen by all cells of "(a)(b)V..." which contain "a" and/or can contain "b" if "ab" is seen by all cells of "(a)(b)V..." which contain "b" => "a" can be removed from "abY". Note that "U", and "V" need not be disjoint.

abZ     ab   (a)(b)V...
abX     abY  (a)(b)U...

UR + 4

UR + 4kx

Four cells, three with one extra candidate, "abx", "abz", and "abw", and one with at least one other extra candidate, "abY" similar to UR+4KX with U, V, and T some appropriate combination of {"(a)(w)(z)x", "(a)(x)(w)z", "(a)(x)(z)w"} => "a" can be removed "abY".

abw     abz  (a)(x)(w)z (a)(x)(z)w
abx     abY  (a)(w)(z)x

UR + 4KX

Four cells with extra candidates "X", "Y", "Z", and "W", plus extra cells, "aU...", such that U is a locked set which includes "X", "abX" is seen by all cells of "aU..." which contain elements of "X", and "abY" is seen by all cells of "aU..." which contain "a", plus additional extra cells "aV...", such that V is a locked set which includes "Z", "abZ" is seen by all cells of "aV..." which contain elements of "Z", and "abY" is seen by all cells of "aV..." which contain "a", plus additional extra cells "aT...", such that "T" is a locked set which includes "W", "abW" is seen by all cells of "aT..." which contain elements of "W", and "abT" is seen by all cells of "abY" which contain "a" => "a" can be removed from "abY". Note that "U", "V", and "T" need not be disjoint.

abW     abZ  aV... aT...
abX     abY  aU...

UR + 4rx

RW has also developed two techniques which use something like an Empty Rectangle.

Two cells in a line and box, with extra candidates, "X" and "Y", plus "b" only exists possibly in "abX" and in line with the rightmost "ab" and "abY" => "a" can be removed from "abY"

ab    ab
----------
abX - abY
 -  - (b)...
 -  - (b)...

UR + 4rd

Two diagonal cells, with extra candidates, "X" and "Y", with "abX" and the rightmost "ab" in a box, plus "a" only exists possibly in "abX" and in line with the rightmost "ab" and "abY" => "a" can be removed from "abY"

ab    abY
----------
abX - ab
 -  - (a)...
 -  - (a)...

Miscellaneous

Here are how RW's later set of ALS examples maps into the above definitions. There are a couple which, as far as I can tell, do not map into ALS and thus not into the definitions. I'm happy to let these go for now and deal with them later.

UR+3KD: X=z, Z=z, U={xz,abx}, V=U
abz    abz    xz
ab     abY    abx
 
UR+3kd:
abz    ab     bz
abx    abY    ax
 
UR+2KD: U={"bxz", "bxz"}
abx     ab     bxz    bxz
ab      abY
 
UR+3KD: X=x, Z=x, U={axz,abz}, V=U
abx    ab
----------
axz    abz
abx    abY
 
UR+3KD: X=x, Z=z, U={bz,xz}, V={bz}
abz    ab     bz
abx    abY    xz
 
UR+2KD: X=z, U={bxz,bx}
abz    ab     bxz
ab     abY    bx
 
Non-ALS configuration:
az
ab     ab
----------------
abxz   abY    ax
 
Non-ALS configuration:
ax   
abz    ab     bz
----------------
abx    abY   
 
UR+3KD: X=xz, Z=xz, U={axz,axz}, V=U
abxz   ab
----------
axz   
axz   
abxz   abY
 
UR+2KX: X=xz, U={abxz,abxz}
ab     abxz
-----------
       abxz
       abxz
ab     abY

Although trying to figure out the general rules has been fun, its time to start programming and see what else I missed.