Unique Rectangles: The Essentials

Advanced methods and approaches for solving Sudoku puzzles

Only one UR

Postby keith » Sun Jul 09, 2006 11:53 am

In your second example, (relabeled):

Code: Select all
*--------------------------------------------------------------------*
| 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      |
*--------------------------------------------------------------------*
 

*49 is a Type 1 UR and you can eliminate <49> in R2C6.

The corners of a UR must be contained within two boxes. The rectangles #29 and @29 are not valid for UR reductions.

Keith
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Postby ravel » Sun Jul 09, 2006 12:57 pm

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

The eliminations bring you here, where the puzzle can be solved with an UR type 1:)
Code: Select all
*-----------------------------------------------------------*
| 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   58    | 2     1     57    |
| 7     68    1     | 69    2     589   |*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     14    45    |*36   *36    15    |
| 1     3     6     | 2     59    7     | 4     59    8     |
*-----------------------------------------------------------*
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Postby daj95376 » Sun Jul 09, 2006 1:29 pm

Thanks keith, ravel, and ronk!!!!!!

~Edit~ Impressive! Three of four URs could be reduced ... and then another UR (Type 1) results and leads to a solution of the puzzle.

I'll review the effect of strong links on URs ... and remind myself that they occur in only two rows/columns/boxes.
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Postby r.e.s. » Sun Jul 09, 2006 8:37 pm

daj95376 wrote:Then I'd know if any of the following four Unique Rectangles can be reduced.

Code: Select all
*-----------------------------------------------------------*
| 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     |
*-----------------------------------------------------------*

This is a good opportunity to make a point about the way candidate grids are often presented -- namely, among the singletons no attempt is made to distinguish the given digits (clues) from the solved digits. This distinction is important when the solving-methods of interest include unique-solution strategies!

E.g, in addition to the UR's already indicated in the above grid, there are many potential unavoidable rectangles from which inferences may be drawn if their "corner singletons" are solved digits and not clues; here are four potential unavoidable rectangles indicated by four different bracket-types:

Code: Select all
*-----------------------------------------------------------*
| 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}   <8>6   9     | 3     567   568   |<2>    1    {7}5   |
|{7}    68    1     | 69    2     5689  | 36    3569 {4}    |
| 5    <2>    3     | 4     679   1     |<8>    69    79    |
|-------------------+-------------------+-------------------|
|[8]    4     5     |[9]6   13    369   | 7     2     139   |
|[9]    7     2     |[8]    134   345   | 136   356   135   |
| 1     3     6     | 2     59    7     |(4)   (5)9    8    |
*-----------------------------------------------------------*

Of these four, it can be concluded (assuming a unique solution) that in the original puzzle at least one of the singletons in the <8>6-<2>-<8>-<2> rectangle was given as a clue, and so was at least one of the singletons in the {4}-{7}-{4}-{7}5 rectangle. Consequently, no unique-solution strategy can be based on these two potential unavoidable rectangles.

OTOH, one can't tell (from the information as-presented) whether some of the singletons in the other two rectangles were given as clues; if none were clues, then [8]-[9]-[8]-[9]6 places that 6, and (5)-(4)-(5)9-(4) places that 9.
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Postby daj95376 » Sun Jul 09, 2006 9:58 pm

r.e.s. ......... Here's the original puzzle.

Code: Select all
*-----------*
|6.8|5..|9..|
|...|..2|..6|
|.54|...|.8.|
|---+---+---|
|...|3..|.1.|
|7..|.2.|..4|
|.2.|..1|...|
|---+---+---|
|.4.|...|72.|
|9..|8..|...|
|..6|..7|4.8|
*-----------*
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Postby daj95376 » Wed Jul 12, 2006 8:44 am

Ignoring my earlier confusion -- which keith finally managed to get through my head -- how about two overlapping URs(ab) with short forcing chains that cause mutual eliminations. I'll throw in UR(c) for free.

Code: Select all
Initial puzzle and reduction to URs.

 *-----------*
 |...|...|769|
 |..7|..4|5..|
 |65.|..7|.4.|
 |---+---+---|
 |.23|7..|..8|
 |..9|5.3|6..|
 |8..|..9|45.|
 |---+---+---|
 |.3.|6..|.74|
 |..1|4..|2..|
 |468|...|...|
 *-----------*

 *-----------------------------------------------------------*
 | 3     18a   4     | 128  -125   125   | 7     6     9     |   just (-1)
 | 9     18a   7     |-138   6     4     | 5     38    2     |   just (-1)
 | 6     5     2     | 389   39    7     | 38    4     1     |
 |-------------------+-------------------+-------------------|
 | 5     2     3     | 7     4     6     | 19c   19c   8     |
 | 1     4     9     | 5     8     3     | 6     2     7     |
 | 8     7     6     | 12b   12b   9     | 4     5     3     |
 |-------------------+-------------------+-------------------|
 | 2     3     5     | 6     19    18    | 189   7     4     |
 | 7     9     1     | 4     35    58    | 2     38    6     |
 | 4     6     8     | 1329  7     12    | 3-19  19c   5     |
 *-----------------------------------------------------------*
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Postby Carcul » Wed Jul 12, 2006 11:23 am

Keith wrote:The corners of a UR must be contained within two boxes.


In general we have:

We have an Unique Pattern on two candidates "x" and "y", if we have 2n cells populated only by "x" and "y" (except at least one of them, which must contain candidates "x", "y", and at least another one), and distributed by exactly n boxes, n rows, and n columns (where n>1).

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Postby Carcul » Wed Jul 12, 2006 11:35 am

Daj95376 wrote:...how about two overlapping URs(ab)...
Code: Select all
*-----------------------------------------------------------*
 | 3     18a   4     | 128  -125   125   | 7     6     9     |   just (-1)
 | 9     18a   7     |-138   6     4     | 5     38    2     |   just (-1)
 | 6     5     2     | 389   39    7     | 38    4     1     |
 |-------------------+-------------------+-------------------|
 | 5     2     3     | 7     4     6     | 19c   19c   8     |
 | 1     4     9     | 5     8     3     | 6     2     7     |
 | 8     7     6     | 12b   12b   9     | 4     5     3     |
 |-------------------+-------------------+-------------------|
 | 2     3     5     | 6     19    18    | 189   7     4     |
 | 7     9     1     | 4     35    58    | 2     38    6     |
 | 4     6     8     | 1329  7     12    | 3-19  19c   5     |
 *-----------------------------------------------------------*


Regarding those two overlapping URs labeled "a" (r12c24) and "b" (r16c45) you could do better then that:

r1c5=5 or r2c4=3, and so, in either case, r3c5<>3 which solve the puzzle.

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Postby daj95376 » Wed Jul 12, 2006 4:43 pm

Carcul wrote:
Keith wrote:The corners of a UR must be contained within two boxes.


In general we have:

We have an Unique Pattern on two candidates "x" and "y", if we have 2n cells populated only by "x" and "y" (except at least one of them, which must contain candidates "x", "y", and at least another one), and distributed by exactly n boxes, n rows, and n columns (where n>1).

Carcul


Very interesting!!! What a nightmare it must be to search for this pattern.
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Postby keith » Wed Jul 12, 2006 10:42 pm

Very interesting!!! What a nightmare it must be to search for this pattern.

Not at all! Here is an example.

Code: Select all
Puzzle: AU070806
+-------+-------+-------+
| . . 8 | . . . | . 2 . |
| . 9 7 | 3 . . | . . . |
| . . . | 5 8 . | 1 . . |
+-------+-------+-------+
| 6 4 . | . . . | 2 . . |
| . . . | . 7 . | . . . |
| . . 2 | . . . | . 5 4 |
+-------+-------+-------+
| . . 5 | . 1 6 | . . . |
| . . . | . . 7 | 8 9 . |
| . 8 . | . . . | 5 . . |
+-------+-------+-------+

The basic techniques get you to here:
Code: Select all
+----------------+----------------+----------------+
| 345  1    8    | 7    46   49   | 3469 2    569  |
| 245  9    7    | 3    246  1    | 46   68   568  |
| 234  236  46   | 5    8    249  | 1    37   79   |
+----------------+----------------+----------------+
| 6    4    39   | 1    39   5    | 2    78   78   |
| 8    5    39   | 24   7    24   | 369  136  169  |
| 1    7    2    | 6    39   8    | 39   5    4    |
+----------------+----------------+----------------+
| 9    23   5    | 8    1    6    | 7    4    23   |
| 234  236  146  | 24   5    7    | 8    9    1236 |
| 7    8    146  | 9    24   3    | 5    16   126  |
+----------------+----------------+----------------+

Note the swordfishy thing on <39> in R456C357. So, R5C7 must be <6>.

I am not pretending these are common, but they are easy to spot.

Keith
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Postby Darkozin » Thu Jul 13, 2006 8:35 pm

Hi Keith

Wow! You did a great job on your guide of Unique Rectangles

Keep up the good work
-Darkozin-
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Postby daj95376 » Sat Jul 22, 2006 5:50 am

Keith, do I have a UR below? If so, what type?

Code: Select all
 *--------------------------------------------------*
 | 2    18   3    | 6    18#  4    | 9    5    7    |
 | 9    4    5    | 7    38#  2    | 36   68   1    |
 | 6    178  78   |*158@ 9   *358@ | 23   28   4    |
 |----------------+----------------+----------------|
 | 3    5    67   | 4    27   1    | 26   9    8    |
 | 1    679  679  |*58   27  *58   | 4    236  23   |
 | 8    2    4    | 3    6    9    | 1    7    5    |
 |----------------+----------------+----------------|
 | 5    89   289  | 18   4    38   | 7    123  6    |
 | 4    68   268  | 9    138@ 7    | 5    123  23   |
 | 7    3    1    | 2    5    6    | 8    4    9    |
 *--------------------------------------------------*

UR (Type ???) in [r35c46] with <58> -- (*). So, we must have [r3c4]=1 or [r3c6]=3. In either case, we're going to end up with [r1c5]=8 or [r2c5]=8 -- (#). This means that [r3c4]<>8, [r3c6]<>8, and [r8c5]<>8 -- (@).
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Postby ravel » Sat Jul 22, 2006 1:43 pm

Since the 5's in row 3 are strongly linked, you have a UR type 4 and can eliminate the both 8's anyway (r8c5 then follows from box/line elimination).
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Postby daj95376 » Sat Jul 22, 2006 5:37 pm

ravel wrote:Since the 5's in row 3 are strongly linked, you have a UR type 4 and can eliminate the both 8's anyway (r8c5 then follows from box/line elimination).

Thanks ravel!!! I knew I was missing something simple in why those 8s could be removed.
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Postby ravel » Sat Jul 22, 2006 9:55 pm

It should be said, that your deduction also would work, when there are other 5's in row 3.
It could be seen as a variant of a type 3 UR. There you have 2 cells with ab only and the extra candidiates in the other 2 cells form an n-tuple with n-1 cells in the same unit - you can eliminate the tuple candidates from the rest of the unit.
In this case a or b is part of the n-tuple. Then you also can safely eliminate a or b from the 2 UR cells.
I dont know, if this variant already has a name.
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