Unique Rectangle Gallery #81
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KyouyamaKazusa0805
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Link: http://forum.enjoysudoku.com/uniqueness-type-6-ur-meets-x-wing-t3709-30.html#p26448
Keith has written an Introduction to Unique Rectangles. Here's a shot at all of the UR options based on Myth's nomenclature. Because the descriptions of the different types are important, I've added some additional labeling which isn't strictly necessary, for example the "X" in UR2X/1SL to distinguish where the strong link is located. In addition, "Type x" refer to the historical typing definitions.
UR + 1 (Type 1)
One UR cell with extra candidates => "ab" can be removed from "abX".
UR + 2
Two UR cells with a extra candidate / extra candidates.
UR + 2x (Type 2/2b)
Two cells with the same one extra candidate in a line => "x" can be removed from all cells common to the "abx".
UR + 2d (Type 5)
Two cells with the extra candidate diagonal to each other => "x" can be removed from all cells common to the "abx" (given the rightmost "abx" and "ab" share a box, then at the "*").
UR + 2X (Type 3/3b)
Two cells with one or more, not necessarily equivalent, extra candidates => treat "abX" as "X+Y" and "abY" as "abX+Y" and perform naked and hidden set reductions in the unit(s) containing these cells except for these cells.
UR + 2D (UR-ALS-XZ)
Two cells with extra candidates diagonal to each other such that "X+Y"="xy" and a naked "xy" common the "abX" and "abY" => eliminate "xy" from the cell common to all three (given "abX" and the rightmost "ab" share a box then at the "*").
UR + 2/1SL
Two UR cells with extra candidates, strong link between two cells.
UR + 2X/1SL (Type 4)
Link between both cells with extra candidates, both cells with extra candidates in a line which forms an X-wing => X-wing eliminations if not previously performed (a strong link on "a" removes "a" in cells common to the rightmost "ab" and "abX" and "a" in cells comon to the leftmost "ab" and "abY"), and "b" from "abX" and "abY"
UR + 2B/1SL
Link between one cell with extra candidates and one bivalue cell, both cells with extra candidates in a line => a strong link on "a" removes "b" from "abY" - repeat for each strong link.
UR + 2D/1SL
Cells with extra candidates diagonal to each other => strong link on "a" removes "a" from "abY" - repeat for each strong link.
UR + 3
Three UR cells with extra candidate(s).
UR + 3x (Type 5)
Three cells with the same one extra candidate => "x" can be removed from all cells common to the "abx" (given the rightmost "abx" share a box, then at the "*").
UR + 3X (UR-ALS-XZ)
Three cells with extra candidates diagonal to each other such that "X+Y+Z"="xy" and a naked "xy" common the "abX", "abY", and "abZ" => eliminate "xy" from the cell common to all four (given "abX" and "abZ" share a box then at the "*").
UR + 3/1SL
Two or three UR cells with extra candidates (Z is optional), plus one strong link and at least one extra cell.
UR + 3x/1SL
"Y" is a single candidate "y", the extra cell "(ab)y" can include "a", and/or can include "b" if it shares a house with "abX" => "b" can be removed from "abX". Similarly, the extra cell "(ab)y" can include "b", and/or can include "a" if it shares a house with "abX" => "b" can be removed from "ab(Z)".
UR + 3X/1SL
Includes the extra cell "(ab)U..." such that "U" is a locked set which includes "Y", "abY" is seen by all of the cells of "(ab)U..." which contain elements of "Y", "(ab)U..." can contain "a", and "(ab)U..." can contain "b" if all of its cells which contain "b" are seen by "abX" => "b" can be removed from "abX". Similarly, "(ab)U..." can contain "b", and "(ab)U..." can contain "a" if all of its cells which contain "b" are seen by "abX" => "b" can be removed from "ab(Z)".
UR + 3/2SL
Three UR cells with extra candidates, plus two strong links, at most one of which includes the bivalue cell.
UR + 3X/2SL
Both strong links share a node, do not include the bivalue cell and have different labels which forms a continuous nice loop => strong links as shown remove "a" in cells common to "ab" and "abY", "b" in cells common to "ab" and "abX", and "Z" in "abZ" which reduces the problem to UR+2D/1SL so "b" can be removed from "abY" and "a" can be removed from "abX".
UR + 3C/2SL
Both strong links share a node, do not include the bivalue cell and have equal labels => "b" can be removed from "abZ".
UR + 3N/2SL
Both strong links share a node, have different labels and one link includes the bivalue cell => "a" can be removed from "abY", and treat "abY" as "Y+Z" and "abZ" as "abY+Z" and perform naked and hidden set reductions in the unit(s) containing these cells except for these cells.
UR + 3U/2SL
The strong links are disjoint with different labels => "a" can be removed from "abY".
UR + 3E/2SL
The strong links are disjoint with the same labels which forms an X-wing => X-wing eliminations if not previously performed (strong links as shown remove "a" from cells common to "ab" and "abY" and "a" in cells common to "abX" and "abZ"), and "b" from "abZ".
UR + 4
All four UR cells with extra candidate(s).
UR + 4/1SL
Four UR cells with extra candidates, plus one strong link and at least two extra cells
UR + 4x/1SL
"Y" and "Z" are single candidates "y" and "z", the extra cell "(ab)y" can contain "a" if it shares a house with "abW" and/or "b" if it shares a house with "abX", similarly the extra cell "(ab)z" can contain "a" if it shares a house with "abW" and/or "b" if it shares a house with "abX" => "b" can be removed from "abX".
UR + 4X/1SL
Includes the extra cell(s) "(ab)U..." such that "U" is a locked set which includes "Y", "abY" is seen by all of the cells of "(ab)U..." which contain elements of "Y", and "(ab)U..." can contain "a" if all of its cells which contain "a" are seen by "abW" and/or can contain "b" if all of its cells which contain "b" are seen by "abX" and similarly for "(ab)V..." where "V" is a locked set which includes "Z" => "b" can be removed from "abX".
UR + 4/2SL
Three or four UR cells with extra candidates (Z is optional), plus two strong link and at least one extra cell.
UR + 4x/2SL
"Y" is a single candidate "y", the extra cell "(ab)y" can include "a", and/or can include "b" if it shares a house with "abX" => "b" can be removed from "abX".
UR + 4X/2SL
Includes the extra cell "(ab)U..." such that "U" is a locked set which includes "Y", "abY" is seen by all of the cells of "(ab)U..." which contain elements of "Y", "(ab)U..." can contain "a", and "(ab)U..." can contain "b" if all of its cells which contain "b" are seen by "abX" => "b" can be removed from "abX".
UR + 4/3SL
All four UR cells with extra candidates, plus three strong links.
UR + 4X/3SL
The links with equal labels are disjoint which forms a continuous nice loop => X-wing eliminations if not previously performed ("a" can be removed from cells common to "abX" and "abY" and cells common to "abZ" and "abW"), "Z" can be removed from "abZ", "W" can be removed from "abW" which reduces the problem to UR+2B/1SL so "b" can be removed from "abX" and "abY".
UR + 4C/3SL
The links with equal labels share a node => "b" can be removed from "abZ".
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