Intersecting UR1's

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Intersecting UR1's

Postby civiliza » Mon Aug 20, 2012 7:27 am

In the puzzle:
Code: Select all
026009000
000000000
500000007
010000006
489010003
000002050
008041090
040000071
090300000


I have reached the point where I seem to have two Intersecting UR1's in the rightmost three columns.

Simplified (with only the possible values for the seven squares showing) this looks like:
Code: Select all
+====+====+====+
|45  |    |45  |
+----+----+----+
|    |    |    |
+----+----+----+
|    |    |    |
+====+====+====+
|4  8|4  8|    |
+----+----+----+
|    |    |    |
+----+----+----+
|    |    |    |
+====+====+====+
|    |    |    |
+----+----+----+
|    |    |    |
+----+----+----+
|4568|4  8|45  |
+====+====+====+


Can the 4568 square safely be reduced to the value 6? The 45 UR removes the 4 and the 5 while the 48 UR removes the 4 and the 8. So 45 and 8 can all be eliminated leaving just the 6, but performing one UR Elimination stops the other working and vice versa.

Just out of curiosity - I've archived the original, so I can revisit it another time.

APOLOGIES - I gave the wrong initial puzzle the first time I posted this.
THANKS/APOLOGIES - As DAJ realised, I also asked about the 3468 square instead of the 4568 square (Gnifer Trouble)
Last edited by civiliza on Mon Aug 20, 2012 4:48 pm, edited 1 time in total.
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Re: Intersecting UR1's

Postby daj95376 » Mon Aug 20, 2012 3:58 pm

civiliza wrote:Can the 4568 square safely be reduced to the value 6?

Yes.
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Re: Intersecting UR1's

Postby civiliza » Mon Aug 20, 2012 4:43 pm

Thank you DAJ.

I had a hunch that the reduction was valid, but having got stuck one step after using it, I thought it better to ask.

I was pleased to note that Sudoku Explainer also reached the point where both UR's were visible because I was worried I had made an earlier mistake. Since I was trying to solve it without SE's hints, I did not use it to investigate the alternatives.

Out of curiosity, does this sort of elimination have a name? My best guess would be some sort of Unique Chain, but since I have deliberately shied away from chains of all sorts that is a stab in the dark. (The shape reminds me of a swordfish - Is the Unique Rectangle the base member of a family of Unique Fish or are fish single value by definition?)
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Re: Intersecting UR1's

Postby JC Van Hay » Mon Aug 20, 2012 5:28 pm

Here are some proofs ...

DP(48)r49c12=[6r9c1 or DP(45)r19c13] :=> +6r9c1
DP(45)r19c13=[6r9c1 or DP(48)r49c12] :=> +6r9c1

Whether r9c2=4 or r9c2=8 -> +6r9c1 to avoid DP(48)r49c12 or DP(45)r19c13
Whether r9c3=4 or r9c3=5 -> +6r9c1 to avoid DP(45)r19c13 or DP(48)r49c12
and the same, mutatis mutandis, with the other bivalues cells containing either 45 or 48.
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Re: Intersecting UR1's

Postby ronk » Mon Aug 20, 2012 7:02 pm

civiliza wrote:Out of curiosity, does this sort of elimination have a name?

No special name. It is merely batch processing of the independent deductions of two URs that happen to share a cell, and happen to share an exclusion.
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Re: Intersecting UR1's

Postby David P Bird » Mon Aug 20, 2012 7:30 pm

civiliza If you make the eliminations for one of the URs you are left with what's called an Avoidable Rectangle for the other. ARs can arise following other types of eliminations too.

The rule for both URs and ARs is that for a rectangle of four cells in two boxes, if none of the cells contains a given, then they must eventually contain at least 3 different digits.
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Re: Intersecting UR1's

Postby civiliza » Tue Aug 21, 2012 11:43 am

Thanks David, I was planning on posting my (non algorithmic) reinterpretation of the implications of JC's proof, but I think that what you said is clearer than anything I could come up with.

I was eventually able to follow the proof, but being shaky on both unique terminology and algorthmic understanding I was having difficulty putting it into words.

Thanks also to JC and to Ronk - After my last post I was wondering if this was a limited form of Binary Universal Graveyard, but I've only just got used to the BUG as a full puzzle issue. I think I saw a post on this site once of a small scale BUG, but at the time I didn't examine it to understand why it was still a BUG.

I'll put it all down to a serendipitous artifact of this particular (randomly generated) puzzle.
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