Question about Unique Rectangles

Advanced methods and approaches for solving Sudoku puzzles

Question about Unique Rectangles

Postby emalvick » Thu Nov 16, 2006 5:56 pm

I am relatively new to the forum, and I am trying to get a grasp on some of the "simpler" advanced techniques that I find useful to solving some of the more challenging puzzles I find by hand.

I recently found the unique rectangles technique, which I find useful for puzzles that I otherwise came to a standstill on.

I understand the different types to a certain extent (as posted in the "essentials" thread/guide). The one thing I haven't been able to find is a discussion into why the URs need to be in only 2 boxes. I know there must be a simple reason, and I am sure there is a discussion in this forum somewhere, but I haven't had luck finding it.

edit: I can see why certain UR types need to be only in 2 boxes, but I am not certain if all of the UR types need to be only in 2 boxes, e.g. Type 1. Couldn't that work in 4 boxes? I'm probably missing something obvious.
:end edit

Could some one post a link and point me in the right direction?

This forum has been invaluable, and I've learned a lot with regard to the UR's and all the FISH methods that have gotten me through some of the more challenging puzzles I hadn't been able to solve by hand before.

Thanks,

Erik
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Re: Question about Unique Rectangles

Postby ronk » Thu Nov 16, 2006 6:37 pm

emalvick wrote:The one thing I haven't been able to find is a discussion into why the URs need to be in only 2 boxes.

Because when you exchange the placement of UR digits, each digit must still appear exactly once in each row, column and box.
Code: Select all
 1 . 2 | . . . | . . .        2 . 1 | . . . | . . .
 . . . | . . . | . . .        . . . | . . . | . . .
 . . . | . . . | . . .        . . . | . . . | . . .
-------+-------+-------      -------+-------+-------
 . . . | . . . | . . .        . . . | . . . | . . .
 2 . 1 | . . . | . . .        1 . 2 | . . . | . . .
 . . . | . . . | . . .        . . . | . . . | . . .
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Re: Question about Unique Rectangles

Postby emalvick » Thu Nov 16, 2006 6:56 pm

I guess what I am hung up on is that why could one use a UR Type 1 for the following situation?

Code: Select all
 12 .  . | .  123 .  | . . .       
 .  .  . |  .  .  .  | . . .       
 .  .  . |  .  .  .  | . . .       
---------+-----------+-------   
 .  .  . |  .  .  .  | . . .       
 12 .  . |  .  12 .  | . . .       
 .  .  . |  .  .  .  | . . .       


It seems like you have to deduct that 12 cannot be in R1C5 even though each corner is in a different box.

I guess it just seems like that for some of the types, that 2 boxes would not necessarily be a requirement.

Perhaps the one thing I am not accounting for then is the other canditates 12 in each box, which could constrain the "UR" from actually being a UR... i.e. if other candidates in a box force one of the corners to take a value that makes the result unique regardless of the corners shown.

For instance, if a candidate in the upper right box in my example forces R1C5 to be 1 or 2, then we aren't left with a deadly combo since the logic led to that answer only.

Anyway, I am sorry to think out loud as I work this out. It makes more sense as I think about it. If my own internal logic is faulty, please let me know. I appreciate all the help I can get.

Erik
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Postby RW » Thu Nov 16, 2006 10:13 pm

emalvick wrote:Perhaps the one thing I am not accounting for then is the other canditates 12 in each box, which could constrain the "UR" from actually being a UR... i.e. if other candidates in a box force one of the corners to take a value that makes the result unique regardless of the corners shown.

That is correct. I suppose this is easiest to demonstrate with real examples. First, try to solve this puzzle with logic only:
Code: Select all
 *-----------*
 |589|241|673|
 |7..|936|548|
 |346|857|192|
 |---+---+---|
 |9..|568|734|
 |875|413|269|
 |463|729|851|
 |---+---+---|
 |694|172|385|
 |257|384|916|
 |138|695|427|
 *-----------*

and you'll find that you cannot, because it has multiple solutions. Next, try this where the rectangle is spread over 4 boxes:
Code: Select all
 *-----------*
 |348|617|295|
 |6.5|.94|387|
 |972|538|416|
 |---+---+---|
 |4.3|.59|678|
 |169|873|524|
 |857|462|931|
 |---+---+---|
 |531|986|742|
 |784|325|169|
 |296|741|853|
 *-----------*

and you'll see that this rectangle doesn't cause multiple solutions.

RW
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Postby emalvick » Thu Nov 16, 2006 11:46 pm

RW wrote:
emalvick wrote:Perhaps the one thing I am not accounting for then is the other canditates 12 in each box, which could constrain the "UR" from actually being a UR... i.e. if other candidates in a box force one of the corners to take a value that makes the result unique regardless of the corners shown.

That is correct. I suppose this is easiest to demonstrate with real examples. First, try to solve this puzzle with logic only:
Code: Select all
 *-----------*
 |589|241|673|
 |7..|936|548|
 |346|857|192|
 |---+---+---|
 |9..|568|734|
 |875|413|269|
 |463|729|851|
 |---+---+---|
 |694|172|385|
 |257|384|916|
 |138|695|427|
 *-----------*

and you'll find that you cannot, because it has multiple solutions. Next, try this where the rectangle is spread over 4 boxes:
Code: Select all
 *-----------*
 |348|617|295|
 |6.5|.94|387|
 |972|538|416|
 |---+---+---|
 |4.3|.59|678|
 |169|873|524|
 |857|462|931|
 |---+---+---|
 |531|986|742|
 |784|325|169|
 |296|741|853|
 *-----------*

and you'll see that this rectangle doesn't cause multiple solutions.

RW


Thank you, as simple as that is, that is what I needed to see. My thoughts were focusing on cases where we know very little about a puzzle, where your examples make it clear by showing the puzzle as nearly complete.

I think by looking at 4 boxes as I was trying, I am neglecting the influence other cells will have and the fact that the unknowns will be part of a bigger/more complex uniqueness condition (i.e. BUG) if it exists.

Working these things out through examples will help me as well.

Thanks,

Erik
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Joined: 01 August 2005

Postby Carcul » Fri Nov 17, 2006 10:43 am

Emalvick wrote:Working these things out through examples will help me as well.


Good idea. Try here for some examples. After that, you may want to check out Almost Unique Rectangles.

Carcul
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Postby emalvick » Fri Nov 17, 2006 9:21 pm

Carcul wrote:Good idea. Try here for some examples. After that, you may want to check out Almost Unique Rectangles.

Carcul


Thanks for the link to the examples. I started with the first example (working by hand) and came up to the following puzzle:

Code: Select all
 *--------------------------------------------------------------------*
 | 7      28     28     | 45     345    35     | 6      1      9      |
 | 4      6      5      | 1      9      7      | 28     28     3      |
 | 139    13     39     | 6      8      2      | 7      5      4      |
 |----------------------+----------------------+----------------------|
 | 1358   9      2348   | 458    12456  1568   | 28     2368   7      |
 | 18     1278   278    | 3      126    9      | 4      268    5      |
 | 358    23458  6      | 7      245    58     | 9      238    1      |
 |----------------------+----------------------+----------------------|
 | 68     47     1      | 9      36     368    | 5      47     2      |
 | 2      358    38     | 58     7      4      | 1      9      6      |
 | 569    457    479    | 2      156    156    | 3      47     8      |
 *--------------------------------------------------------------------*


Now in looking for UR's, and I see two type 1's:

<28> in R24C78 and <47> in R79C28 that let me make some reductions. The puzzle might fall out from that, but I am trying to learn this method. My question is in the potential UR on <28> in R15C23. This looks similar to a type 5 UR, and considering the <18> in R5C1, it seems like I should be able to make some deductions.

However, I don't see it. Am I right in thinking that the 7 (which can only be in R5C2 or R5C3 basically eliminates the possibility of the deadly condition? As a result I really can't make any reductions based on this "potential" UR?

Thanks for the help on this.

Erik
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Postby Carcul » Fri Nov 17, 2006 10:23 pm

Emalvick wrote:As a result I really can't make any reductions based on this "potential" UR?


I don't see any, because "7" must be in one of the cells r5c23, and this candidate doesn't belong to the "deadly candidates" "2,8". If the situation were "r5c2 or r5c3 must be 8" or ""r5c2 or r5c3 must be 2", then we would be able to deduce that r5c23<>2 or r5c23<>8, respectively.

Carcul
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