## Type 3 Unique Rectangles - Hidden Subsets?

Advanced methods and approaches for solving Sudoku puzzles

### Another example

Do you guys have this one cataloged? This fragment R6789 was posted on the Daily Sudoku site, by a human solver. From the initial puzzle, it is not a situation most would reach. Anyway:

Code: Select all
`|1257  237   8     |1257  1267  4     |3679  39    26    | ---------------------------------------------------------- |6     12    259   |124   1234  1235  |8     29    7     | |3     1278  27    |1278  9     1278  |46    5     46    | |25789 278   4     |2578  2378  6     |39    239   1     | ----------------------------------------------------------`

The UR / AUR <39> is in R69C78. The fifth cell is R7C8, <29>. My elimination would be R6C7 is not <3>. Myth Jellies pointed out the strong link on <9> in R6: R9C8 is not <9>.

(I think I have this right!)

Keith
keith
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According to your candidate grid there would also be a strong link on <3> in box 9 and you could eliminate '3' from r6c7 without the fifth cell.

RW
RW
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My results with Kieth's techniques have been more favorable than Ron's. In the top1465, I'm able to solve 4 more puzzles with these techniques than without (#306, 447, 704, and 1145). Given that these puzzles were not solved even with grouped nice loops, that's pretty impressive. I did not see any puzzles that were previously solved, become unsolved with the addition of these techniques (that doesn't mean it can't happen - I just didn't see it happen). Of the four only in one case (#704) was the puzzle not solved without the techniques and then with the addition of the techniques solved (the other cases required the UR to occur prior to reaching a dead end). The PMs for the unsolved grid are shown below as well as the two UR's which lead to cracking the puzzle (both of which have 3 non-bivalued cells in the UR. The UR is indicated by the "*" and "-" where the elimination occurs at the "-". It requires 2 ALS indicated by the "#" and the "@". I'd be interested in other methods to advance the puzzle from the first grid.

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`UR+3KX: r46c36, r5c123, r4c9 => r4c3<>6+--------------------+--------------------+------------------+|    69  2679    267 |     5   269      3 |   48  148     14 ||  1389   349    348 |    79   489  14789 |    5    6      2 ||   168  2456  24568 |    28  2468    148 |    9    7      3 |+--------------------+--------------------+------------------+|     5     1  -2369 |     4   239   *679 |  268   38    @67 ||   #36 #2346  #2346 |  2378     5    678 |    1    9    467 ||     7     8 *23469 |   239     1    *69 |  246  345     45 |+--------------------+--------------------+------------------+|     2   359      1 |     6   349     49 |    7   45      8 ||     4    69     68 |    18     7      5 |    3    2    169 ||  3689  3579   3578 |  1389  3489      2 |   46  145  14569 |+--------------------+--------------------+------------------+UR+3KX: r45c69, r6c6, r5c123 => r5c6<>6+--------------------+--------------------+------------------+|    69  2679    267 |     5   269      3 |   48  148     14 ||  1389   349    348 |    79   489  14789 |    5    6      2 ||   168  2456  24568 |    28  2468    148 |    9    7      3 |+--------------------+--------------------+------------------+|     5     1    239 |     4   239   *679 |  268   38    *67 ||   @36 @2346  @2346 |  2378     5   -678 |    1    9   *467 ||     7     8  23469 |   239     1    #69 |  246  345     45 |+--------------------+--------------------+------------------+|     2   359      1 |     6   349     49 |    7   45      8 ||     4    69     68 |    18     7      5 |    3    2    169 ||  3689  3579   3578 |  1389  3489      2 |   46  145  14569 |+--------------------+--------------------+------------------+`
Mike Barker

Posts: 458
Joined: 22 January 2006

Mike Barker wrote:In the top1465, I'm able to solve 4 more puzzles with these techniques than without (#306, 447, 704, and 1145).
(...)
Of the four only in one case (#704) was the puzzle not solved without the techniques and then with the addition of the techniques solved (the other cases required the UR to occur prior to reaching a dead end).

ronk
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Location: Southeastern USA

Four more puzzles were solved with Keith's UR technique than without. Of the 4, 3 required the UR to occur in the middle of the solving steps. If the technique was not used, the puzzle was not solved. The UR could not be used at this point because it not longer existed. In the one shown, the puzzle was first solved without using the technique which allows a look at the unsolved puzzle. The puzzle was then solved, by first identifying the two UR's using Keith's technique and then the rest of my solver's techniques. I thought this was useful in that it shows a case where of the techniques my solver uses (naked sets, fish, strong links, XY cycles and other bivalue techniques, UR techniques, BUG-lite, basic and grouped nice loops, and ALS), the only one that seemed to work was Keith's UR. Now I could have an error in my solver, it could be I'm not using big enough ALS or nice loops, or I could be missing a technique and I'd love feedback on this. Barring a mistake, however, this is a great example that the technique is effective.
Mike Barker

Posts: 458
Joined: 22 January 2006

Mike Barker wrote:Of the 4, 3 required the UR to occur in the middle of the solving steps. If the technique was not used, the puzzle was not solved. The UR could not be used at this point because it no longer existed.

Thanks for the clarification. When keith's technique is applied later to #704, is it possible to identify the key step that "destroys the UR"?
ronk
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Location: Southeastern USA

Actually the beauty of 704 is that of the 4 puzzles, it is the only one where the UR is not distroyed by effectively implementing Keith's techniques after everything else. This is the board after all other methods are exhausted showing one of the two UR+3KX which allow the puzzle to be solved:
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`+--------------------+--------------------+------------------+ |    69  2679    267 |     5   269      3 |   48  148     14 | |  1389   349    348 |    79   489  14789 |    5    6      2 | |   168  2456  24568 |    28  2468    148 |    9    7      3 | +--------------------+--------------------+------------------+ |     5     1  -2369 |     4   239   *679 |  268   38    @67 | |   #36 #2346  #2346 |  2378     5    678 |    1    9    467 | |     7     8 *23469 |   239     1    *69 |  246  345     45 | +--------------------+--------------------+------------------+ |     2   359      1 |     6   349     49 |    7   45      8 | |     4    69     68 |    18     7      5 |    3    2    169 | |  3689  3579   3578 |  1389  3489      2 |   46  145  14569 | +--------------------+--------------------+------------------+ `

I went back and looked at #306 where appling Keith's techniques after everything else doesn't lead to a solution, but allowing these eliminations prior to nice loops and ALS does. The two method summaries follow:
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`Without Keith's Techniques:Hidden Single: r1c1 => r1c1=9,r1c56<>9,r78c1<>9Hidden Single: r6c3 => r6c3=2,r6c78<>2Hidden Single: r7c3 => r7c3=8,r78c2<>8,r7c45<>8Hidden Single: r1c9 => r1c9=8Locked Row line/box: r2c12 => r2c4579<>1Naked Single: r2c4 => r2c125<>4,r57c4<>4,r13c5<>4,r13c6<>4Locked Column line/box: r78c2 => r2c2<>7Naked Row Pair: r2c12 => r2c79<>5Naked Block Pair: r2c12 => r1c3<>5Naked Column Pair: r13c3 => r49c3<>4Locked Column line/box: r13c5 => r789c5<>6Locked Column box/box: r139c56 => r456c5<>1,r4c6<>1Hidden Column Pair: r78c2 => r7c2=79,r8c2=79Column X-Wing Fillet-o-Fish: r47c1|r479c6 => r7c5<>4Column X-Wing Fillet-o-Fish: r49c3|r4789c6 => r9c5<>3Column X-Wing Fillet-o-Fish: r49c3|r47c9 => r9c8<>5Column X-Wing Fillet-o-Fish: r49c3|r47c9 => r7c1<>5XYZ-wing: r6c1|r4c3, r2c1 => r4c1<>5UR+2X/1SL: r13c38 => r13c8<>7UR+3U/2SL: r56c24 => r5c2<>1UR+4C/3SL: r37c56 => r7c6<>9ALS xy-rule with B=2 cells: r9c2368-1-r2c5|r1c6-2-r78c4|r8c56|r7c5 => r9c5<>5Locked Row box/box: r7c579|r8c58 => r8c1<>5XY-wing: r8c16|r7c4 => r8c4<>6,r7c1<>6Hidden Single: r7c4 => r7c4=6,r7c79<>6Locked Column line/box: r89c8 => r135c8<>6WXYZ-wing: r3c8|r2c79, r3c3 => r3c79<>7*** UR+2B/1SL: r26c12 => r6c2<>1 ***Nice Loop: r6c8=7=r8c8-7-r8c2-9-r8c4=9=r6c4-9-r6c7=9=r4c7~9~r6c8 => r4c7<>7Nice Loop: r1c7=5=r1c8-5-r8c8=5=r8c5=8=r8c4=9=r6c4-9-r6c7=9=r4c7~9~r1c7 => r4c7<>5Nice Loop: r5c7=2=r5c8-2-r3c8-4-r1c8-5-r8c8=5=r8c5=8=r8c4-8-r5c4~1~r5c7 => r5c7<>1WXYZ-wing: r5c789, r3c9 => r4c9<>6Nice Loop: r4c7=9=r6c7-9-r6c4=9=r8c4=8=r8c5=5=r7c5-5-r7c9=5=r4c9-5-r4c3~3~r4c7 => r4c7<>3Grouped Nice Loop: ALS:r27c9-5-ALS:r8c12468-8-ALS:r5c4789~3~ => r4c9<>3Grouped Nice Loop: r456c5=3=r4c6-3-r4c3-5-r4c9=5=r7c9-5-r7c5=5=r8c5~5~r456c5 => r8c5<>3Incomplete solution`

Code: Select all
`With Keith's Techniques (the UR+2kx is highlighted), the UR which distroys the UR+2kx is highlighted in the previous list):Hidden Single: r1c1 => r1c1=9,r1c56<>9,r78c1<>9Hidden Single: r6c3 => r6c3=2,r6c78<>2Hidden Single: r7c3 => r7c3=8,r78c2<>8,r7c45<>8Hidden Single: r1c9 => r1c9=8Locked Row line/box: r2c12 => r2c4579<>1Naked Single: r2c4 => r2c125<>4,r57c4<>4,r13c5<>4,r13c6<>4Locked Column line/box: r78c2 => r2c2<>7Naked Row Pair: r2c12 => r2c79<>5Naked Block Pair: r2c12 => r1c3<>5Naked Column Pair: r13c3 => r49c3<>4Locked Column line/box: r13c5 => r789c5<>6Locked Column box/box: r139c56 => r456c5<>1,r4c6<>1Hidden Column Pair: r78c2 => r7c2=79,r8c2=79Column X-Wing Fillet-o-Fish: r47c1|r479c6 => r7c5<>4Column X-Wing Fillet-o-Fish: r49c3|r4789c6 => r9c5<>3Column X-Wing Fillet-o-Fish: r49c3|r47c9 => r9c8<>5Column X-Wing Fillet-o-Fish: r49c3|r47c9 => r7c1<>5XYZ-wing: r6c1|r4c3, r2c1 => r4c1<>5*** UR+2kx: r26c12, r4c3 => r6c2<>5 ***UR+2rd: r56c24 => r6c4<>8,r5c2<>1UR+2X/1SL: r13c38 => r13c8<>7UR+4C/3SL: r37c56 => r7c6<>9Grouped Nice Loop: r4c12=1=r6c12-1-r6c4-9-r4c56=9=r4c7~9~r4c12 => r4c7<>1Grouped Nice Loop: ALS:r9c2368-1-ALS:r1c6|r2c5-2-ALS:r78c5|r8c46|r7c4~5~ => r9c5<>5Locked Row box/box: r7c579|r8c58 => r8c1<>5XY-wing: r8c16|r7c4 => r8c4<>6,r7c1<>6Hidden Single: r7c4 => r7c4=6,r7c79<>6Locked Column line/box: r89c8 => r135c8<>6WXYZ-wing: r3c8|r2c79, r3c3 => r3c79<>7Nice Loop: r6c8=7=r8c8-7-r8c2-9-r8c4=9=r6c4-9-r6c7=9=r4c7~9~r6c8 => r4c7<>7Nice Loop: r1c7=5=r1c8-5-r8c8=5=r8c5=8=r8c4=9=r6c4-9-r6c7=9=r4c7~9~r1c7 => r4c7<>5Nice Loop: r5c7=2=r5c8-2-r3c8-4-r1c8-5-r8c8=5=r8c5=8=r8c4-8-r5c4~1~r5c7 => r5c7<>1WXYZ-wing: r5c789, r3c9 => r4c9<>6Nice Loop: r4c7=9=r6c7-9-r6c4=9=r8c4=8=r8c5=5=r7c5-5-r7c9=5=r4c9-5-r4c3~3~r4c7 => r4c7<>3Grouped Nice Loop: ALS:r27c9-5-ALS:r8c12468-8-ALS:r5c4789~3~ => r4c9<>3Grouped Nice Loop: r4c12=1=r4c9=5=r6c78-5-r6c1=5=r2c1=1=r2c2~1~ => r6c2<>1Naked Single: r6c2 => r6c5<>8,r5c2<>8Nice Loop: r7c6=2=r7c5=5=r8c5=8=r5c5=4=r5c2-4-r4c1=4=r7c1~4~r7c6 => r7c6<>4Hidden Single: r7c1 => r7c1=4,r4c1<>4,r9c2<>4Naked Row Triple: r9c238 => r9c6<>3Nice Loop: r9c8=3=r9c3=5=r4c3-5-r4c9=5=r7c9~5~r9c8 => r7c9<>3Nice Loop: r8c1=3=r9c3=5=r4c3-5-r4c9=5=r7c9-5-r7c5=5=r8c5~5~r8c1 => r8c5<>3Nice Loop: r7c9=5=r4c9-5-r4c3=5=r9c3=3=r8c1=6=r8c8~6~r7c9 => r8c8<>5Hidden Single: r8c5 => r8c5=5,r7c5<>5Hidden Single: r8c4 => r8c4=8,r5c4<>8Naked Single: r5c4 => r5c9<>1,r6c4<>1Naked Single: r6c4 => r6c57<>9,r4c56<>9Hidden Single: r5c5 => r5c5=8Hidden Single: r5c2 => r5c2=4,r4c2<>4Hidden Single: r4c7 => r4c7=9Locked Row box/box: r4c1356|r6c15 => r6c78<>3Nice Loop: r7c5=9=r8c6-9-r8c2-7-r8c8=7=r6c8-7-r6c5~3~r7c5 => r7c5<>3Locked Column line/box: r78c6 => r4c6<>3Naked Column Triple: r149c6 => r3c6<>17Nice Loop: r2c7=2=r2c5-2-r3c6=2=r7c6=3=r7c7~3~r2c7 => r2c7<>3Hidden Single: r2c9 => r2c9=3,r5c9<>3Naked Single: r5c9 => r5c7<>6,r3c9<>6Naked Single: r3c9 => r3c57<>1,r4c9<>1,r1c7<>1Hidden Single: r6c7 => r6c7=1,r6c1<>1Naked Block Pair: r6c1|r4c3 => r4c1<>3,r4c2<>5Locked Column line/box: r12c7 => r7c7<>7BUG+3C/2SL (2 buglets): r123c57 => r1c5<>7Nice Loop: r8c1=3=r6c1=5=r6c8=7=r8c8~7~r8c1 => r8c8<>3Nice Loop: r5c7-3-r7c7=3=r7c6=2=r3c6~2~ => r3c7<>2The Solution is completed with singles`
Mike Barker

Posts: 458
Joined: 22 January 2006

Mike Barker wrote:With Keith's Techniques (the UR+2kx is highlighted), the UR which distroys the UR+2kx is highlighted in the previous list) ...

That looks like the difference might be due to the relative order of Keith's technique with other uniqueness techniques ... and NOT necessarily with other non-uniqueness techniques.
ronk
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### A claim

I am going to make a claim which may be difficult for me to prove, if I am challenged to prove it:

I have posted at least two examples of overlapping Unique Rectangles: If you make the reductions for one of the rectangles, you destroy the pattern and reductions of the other. Sequence matters.

My claim: I have seen many examples of overlapping XY-wings, in which the reductions of one XY-wing destroy the other wings and their reductions. Sequence matters.

Sequence matters, for all techniques, not only for Uniqueness techniques.

Keith
keith
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keith wrote:
Code: Select all
`|1257  237   8     |1257  1267  4     |3679  39    26    | ---------------------------------------------------------- |6     12    259   |124   1234  1235  |8     29    7     | |3     1278  27    |1278  9     1278  |46    5     46    | |25789 278   4     |2578  2378  6     |39    239   1     | ----------------------------------------------------------  `

The UR / AUR <39> is in R69C78. The fifth cell is R7C8, <29>. My elimination would be R6C7 is not <3>. Myth Jellies pointed out the strong link on <9> in R6: R9C8 is not <9>.

It took me awhile to see that 9 elimination, but wow, what a great deduction!

RW wrote:According to your candidate grid there would also be a strong link on <3> in box 9 and you could eliminate '3' from r6c7 without the fifth cell.

There doesn't appear to be a strong link on 3 in box 9(the lower right hand one right?).

keith wrote:My claim: I have seen many examples of overlapping XY-wings, in which the reductions of one XY-wing destroy the other wings and their reductions. Sequence matters.

I wouldn't think that sequence should matter. As long as an elimination is valid, it shouldn't matter when you do it. If you identify 2 xy-wings and one destroys the other, the conclusions of both should still be valid. An xy-wing is an xy-wing. I have another claim: The deduction from any solving method should be valid even if it uses 'destroyed' candidates. Now I may be jumping the gun on this, but it's just a gut feeling. We use incorrect candidates in chains all the time. In essence those candidates are not really there, so eliminating one may 'destroy' the loop, but just write it back in and the loop is there. Is there some proven rule that loops only work after all possible simpler reductions have been made... singles, pairs, etc? Hell, I think you should be able to use solved cells in loops, just write in some candidates, but this would probably not be fruitful.

Could you post an example of overlapping xy-wings where one destoys the other and its conclusions? I would be really curious to look into that.
doduff

Posts: 32
Joined: 29 May 2006

doduff wrote:There doesn't appear to be a strong link on 3 in box 9(the lower right hand one right?).

There is, you have only the 2 3's in box 9 (and the one in r9c5 can be eliminated).
ravel

Posts: 998
Joined: 21 February 2006

Ah... I see... locked candidates. cool.

Can someone point me to a proof of why a unique puzzle cannot end up in this state:
Code: Select all
`|1257  27    8     |1257  1267  4     |9     3     26    | ---------------------------------------------------------- |6     12    259   |124   1234  1235  |8     29    7     | |3     1278  27    |1278  9     1278  |46    5     46    | |25789 278   4     |2578  278   6     |3     29    1     | ----------------------------------------------------------`

...fixing the 9 at r6c7.
Basically I want to know why

a b
----
b a

It seems like there could be a puzzle where you get to the state:
a b
----
b ax

and the you want to remove the a in ax by an AUR rule, but what says the x cannot be removed by some chain?
doduff

Posts: 32
Joined: 29 May 2006

### Overlapping XY-wings

doduff wrote:

Could you post an example of overlapping xy-wings where one destoys the other and its conclusions? I would be really curious to look into that.

Here is an example from this past week, in another thread:

http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=935

Keith
keith
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That is an interesting example, along with the remote pairs that come afterward. I thought you meant that one of the xy-wings is actually invalid due to the presence of the other. Both conclusions are still true.

I still think it shouldn't matter what order you apply methods in.
doduff

Posts: 32
Joined: 29 May 2006

### Re:

In May 2006, Mike Barker here wrote:Obviously the definitions can apply with "a" and "b" switched or with "x", "Y", etc switched.

--- 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 â€œaâ€ 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`

Since borders between boxes are not shown, that appears to be quite a mouthful. It presumably covers these six non-isomorphic cases.

Code: Select all
` .  ab . | ab  .  .  | .  .   .       . ab  . | abx .  .  | .  .  .    .  .  . | .   .  .  | .  .   .       .  .  . |  .  .  .  | .  .  .    . abx . | abY .  .  | .  ax  .       . ab  . | abY .  .  | .  .  .   ---------+-----------+-----------    ----- ---+-----------+---------- .  .  . | .   .  .  | .  .   .       .  .  . | ax  .  .  | .  .  .   Candidate "a" may be eliminated from the "abY" cell.                                .  ab . | ab  .   .   | .  .  .      .  ab . | abY .   .   | .  .  .   .  .  . | .   .   .   | .  .  .      .  .  . | .   .   .   | .  .  .   . abY . | abx .   ax  | .  .  .      .  ab . | abx .   ax  | .  .  .   ---------+-------------+---------    ---------+-------------+--------- .  .  . | .   .   .   | .  .  .      .  .  . | .   .   .   | .  .  .  Candidate "a" may be eliminated from the "abY" cell. .  ab . | ab  .   .   | .  .  .      .  ab . | abx .   .   | .  .  .   .  .  . | .   .   .   | .  .  .      .  .  . | .   .   .   | .  .  .   . abx . | abY . a(b)x | .  .  .      .  ab . | abY . a(b)x | .  .  .   ---------+-------------+---------    ---------+-------------+--------- .  .  . | .   .   .   | .  .  .      .  .  . | .   .   .   | .  .  .  Candidates "a" and "b" may both be eliminated from the "abY" cell. Candidate "(b)" may be missing.`

Before I continue with the others in that post, would someone please confirm? daj95376?
ronk
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