## Unique Rectangles: The Essentials

Advanced methods and approaches for solving Sudoku puzzles

### Re: The Guide

daj95376 wrote:I'm confused by the deadly pattern paragraph in the above quote. I don't see it.

I recommend the following POV.
Code: Select all
`+--------------+ |  -   -    -  | | 12   -   123 | |  -   -    -  | +--------------+ | 12   -   124 | |  -   -    -  | |  -   -   13  | +--------------+ `

If there is to be a unique solution -- assuming the upper left cell is r1c1 -- at least one of r2c3=3 or r4c3=4 must be true. Therefore ...

r4c3=1 -> r4c3<>4 -> r2c3=3 -> r6c3<>3 -> r6c3=1 -> r4c3<>1, which implies r4c3<>1

In nice loop terms: r4c3=4|3=r2c3-3-r6c3-1-r4c3, implying r4c3<>1
ronk
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ronk,

I don't contest keith's assertion that the lower right cell of the rectangle should not be <1>. In fact, I agreed with his second POV ... and yours!

What I don't see is how he gets away with calling it a deadly pattern and conveniently voids the lower left cell of <2> in his reduction.

daj95376
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daj95376 wrote:What I don't see is how [edit: keith ...] voids the lower left cell of <2> in his reduction.

He didn't say that.
ronk
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ronk wrote:
daj95376 wrote:What I don't see is how [edit: keith ...] voids the lower left cell of <2> in his reduction.

He didn't say that.

He did in his reply to my questioning the existence of a deadly pattern. (See four messages back in this thread.) All I see is ...

Code: Select all
`+--------------+|  -   -    -  || 12   -   123 ||  -   -    -  |+--------------+| 12   -   124 ||  -   -    -  ||  -   -   13  |+--------------+`

Code: Select all
`+--------------+|  -   -    -  ||  1   -    2  ||  -   -    -  |+--------------+|  2   -    1  ||  -   -    -  ||  -   -    3  |+--------------+`

when you make the (false) premise that the <124> cell has a value of <1> and get (what appears to be) perfectly valid assignments in all five cells.
daj95376
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### Let me try this ...

Here is my reasoning. I am pretty sure it's ok:
Code: Select all
`+--------------+ |  -   -    -  | | 12a  -   123c| |  -   -    -  | +--------------+ | 12b  -   124d| |  -   -    -  | |  -   -   13e | +--------------+ `

Assume d is <1>. Then you can sequentially say: e is <3>, c is <2>, a is <1>, and b is <2>.

Keith
keith
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### Re: Let me try this ...

keith wrote:
Code: Select all
`+--------------+ |  -   -    -  | | 12a  -   123c| |  -   -    -  | +--------------+ | 12b  -   124d| |  -   -    -  | |  -   -   13e | +--------------+ `

Assume d is <1>. Then you can sequentially say: e is <3>, c is <2>, a is <1>, and b is <2>.

Assume d is <2>. Then because of the UR, since d is not <4>, c must be <3>. Also b is <1> which leaves no candidates for e. Therefore, d cannot be <2>.

Whether the extra cell e contains <13> or <23>, it appears the UR may be treated as a Type 1 UR, i.e., as if the <3> didn't even exist in the c cell. And as for the Type 1 UR, the d cell may any number of extra candidates.

Mike Barker, if this one's already in your [edit: master UR] list, I can't find it.
Last edited by ronk on Wed Jun 21, 2006 3:53 pm, edited 1 time in total.
ronk
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I give up. [Edited] Everyone is so fixated on proving that cell <124> can't be <1|2> that they're missing my point. [end-edit] Later!
Last edited by daj95376 on Wed Jun 21, 2006 4:41 pm, edited 3 times in total.
daj95376
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daj95376 wrote:placing a <1> in the <124> cell destroys the deadly pattern and replaces it with an invalid puzzle solution

As you said:invalid solution, that's why we can eliminate <1> from the cell <124>. Mostly when we work on a puzzle we know that there is a valid (unique) solution, the solution we would reach with <1> in <124> would not be unique, therefore that cannot be part of the solution.

ronk wrote:Mike Barker, if this one's already in your list, I can't find it.

Mike Barker wrote:--- 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".
Code: Select all
`ab     ab          abx    abY  (b)(a)x `

Here you could add: if (b)(a)x is common to the "ab" which is in line with "abY" => "b" can be removed from "abY"

In this particular case there also seems to be a x-wing on <2> that would let us eliminate <1> from <123> without considering the extra cell <13>.

RW
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Code: Select all
`+--------------+ |  -   -    -  | | 12   -  B123 | |  -   -    -  | +--------------+ |C12   -  A124 | |  -   -    -  | |  -   -  C13  | +--------------+ `

Here is an AIC that might help. Noting the ALS in set C

A4 =UR= B3 - C3 = C(1&2)

Thus, A = 4 and/or C = (1&2). Either way you can remove the 1 and 2 from A. You can replace 4 in the grid with the more generic X as well.
Myth Jellies

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And here is the corresponding nice loop that uses the link in the AUR:

Code: Select all
`+--------------+ |  -   -    -  | | 12   -  B123 | |  -   -    -  | +--------------+ |C12   -  A124 | |  -   -    -  | |  -   -  C13  | +--------------+ `

[A]=4|3=[B]-3-[C]-1,2-[A], => A<>1,2 => A=4.

Carcul
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daj95376 wrote:Everyone is so fixated on proving that cell <124> can't be <1> that they're missing my point.

Not true. If you'll read my post again, you'll see it shows that cell <124> can't be <2>. That's the lower right cell ... not the lower left cell as you posted.
ronk
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Sorry, I was not able to post this sooner.

The pattern in question is the same one that started the following thread:

http://forum.enjoysudoku.com/viewtopic.php?t=4088&start=0

I think Mike Barker, Havard and others did a great job of classifying and explaining the various patterns and their reductions.

Having had some time to read, and think about, all the messages, I think we all are agreeing. Please consider that I am not trying to break any new ground here, I was trying to help people by documenting some newer techniques.

The point of this part of my UR guide was (and is) the following: Given a pattern like:
Code: Select all
`+--------------+ |  -   -    -  | | 12a  -   123c| |  -   -    -  | +--------------+ | 12b  -   124d| |  -   -    -  | |  -   -   13e | +--------------+ `

you can make a reduction with reasoning based only on the candidates in the given cells.

To remark on one message: RW is exactly correct. If you delete cell "e" and substitute a strong link for <1> (or <2>) in ac (or bd) there is still an elimination of <1> or <2> in c or d.

I am concluding that this is a very common pattern, although the reduction it affords is almost never a puzzle-breaker.

Best wishes,

Keith
keith
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keith,

I wish that I had studied your examples a little closer. Then I'd know if any of the following four Unique Rectangles can be reduced.

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`*-----------------------------------------------------------*| 6     1     8     | 5    *34   *34    | 9     7     2     || 3     9     7     | 1     8     2     | 5     4     6     || 2     5     4     | 7     69    69    |*13    8    *13    ||-------------------+-------------------+-------------------|| 4    *68    9     | 3     567   568   | 2     1     57    || 7    *68    1     |*69    2     5689  | 36    3569  4     || 5     2     3     | 4     679   1     | 8     69    79    ||-------------------+-------------------+-------------------|| 8     4     5     |*69    13    369   | 7     2     139   || 9     7     2     | 8     134   345   | 136   356   135   || 1     3     6     | 2     59    7     | 4     59    8     |*-----------------------------------------------------------*`

Fortunately, two of the three Unique Rectangles in the following puzzle are Type 1 ... and I can reduce them.

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`*--------------------------------------------------------------------*| 7      8      6      |*29     13     13     |*29     5      4      ||*49     1      3      | 24589  458    2459   | 6      7     *29     ||*49     5      2      | 7      6     *49     | 1      3      8      ||----------------------+----------------------+----------------------|| 2      3      7      | 1      45     45     | 8      9      6      || 5      6      9      | 3      2      8      | 4      1      7      || 1      4      8      | 6      9      7      | 3      2      5      ||----------------------+----------------------+----------------------|| 3      279    1      | 289    78     6      | 5      4     *29     || 6      279    5      | 249    347    2349   |*29     8      1      || 8      29     4      | 259    15     1259   | 7      6      3      |*--------------------------------------------------------------------*`
daj95376
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Because there is a strong link for 4 in r8, you can eliminate 3 in r8c56 (UR 34 in r18c56).
Because there is a strong link for 8 in c6, you can eliminate 6 in r45c6 (UR 68 in r45c26).
ravel

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And ... because there is a strong link for 1 in c7, you can eliminate 3 in r8c9 (UR 13 in r38c79).
ronk
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