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What does this condition imply?

 
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tlanglet



Joined: 17 Oct 2007
Posts: 2461
Location: Northern California Foothills

PostPosted: Fri Jul 03, 2009 10:33 pm    Post subject: What does this condition imply? Reply with quote

The following is the code for a previously posted puzzle after basics have been completed.
Code:


 *--------------------------------------------------------------------*
 | 17     9      8      | 2      6      3      | 57     4      15     |
 | 2      4      36     | 1      5      7      | 36     9      8      |
 | 167    367    5      | 8      4      9      | 2      37     16     |
 |----------------------+----------------------+----------------------|
 | 9      67    *267    | 5      3      1      | 8     *26     4      |
 | 8      1      4      | 6      7      2      | 9      5      3      |
 | 5      36    *236    | 9      8      4      | 1     *26     7      |
 |----------------------+----------------------+----------------------|
 | 467    2      1      | 47     9      8      | 34567  37     56     |
 | 3      8      67     | 47     2      5      | 467    1      9      |
 | 47     5      9      | 3      1      6      | 47     8      2      |
 *--------------------------------------------------------------------*

This issue is the UR 26 in r46c38. To prevent the UR, either r4c3=7 or r6c3=3. What are the implications?

If r4c3=7, then r8c3=6, which makes r8c7=47. This forms a subset 47 with cell r9c7 that deletes the 7 in r1c7 making it a 5.

OK, so what about the 3 in r6c3?

If r6c3=3 then r2c3=6, r2c7=3, then r3c8=7, and r1c7=5.

At this point, everything looks simple and straight forward. Set r1c7=5 and the puzzle is completed in one step.

BUT WAIT! Here is another deal

If r4c3=7, r8c3=6, r2c3=3, r2c7=6, r3c9=1, and r1c9=5.

CONFLICT Exclamation Exclamation

From this, I conclude that r4c3=7 produces a conflict and therefore is not valid. Thus the correct move is to eliminate 7 from r4c3 forcing r6c3=3 to prevent the UR condition.

Comments please.

Ted
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sat Jul 04, 2009 3:16 am    Post subject: Re: What does this condition imply? Reply with quote

tlanglet wrote:
Comments please.

Logic comes in many flavors. You are mixing two flavors in one discussion.

Flavor #1: At least one of two conditions must be true. It's possible that they both may be true, but that's seldom the case. All that's necessary is to find a common downstream condition that's true for a chain originating from each. Using the two conditions [r4c3]=7 and/or [r6c3]=3, you found a chain for each with a common outcome; i.e., [r1c7]=5.

Flavor #2: If two chains from the same assumption give conflicting results, then the original assumption is incorrect. This is presented in Jeff's thread on forcing chains. Using two chains based on the assumption [r4c3]=7, you found a contradiction and can thus invalidate the original assumption.

Both conclusions are correct.

Regards, Danny

[Edit: content in red is based on limited personal experience and no facts. Sorry!!!]


Last edited by daj95376 on Thu Jul 09, 2009 5:18 pm; edited 1 time in total
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tlanglet



Joined: 17 Oct 2007
Posts: 2461
Location: Northern California Foothills

PostPosted: Sat Jul 04, 2009 3:28 am    Post subject: Reply with quote

Wunderbar Exclamation

I love it when you do not have to make a decision which way is correct since both are valid.

Thanks Danny.......

Ted
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Wed Jul 08, 2009 10:22 pm    Post subject: Reply with quote

Quote:
It's possible that they both may be true, but that's seldom the case.

Really? I thought it is quite common.

I think the simplest case of "one, or both, is true" is the pincers of an XY-wing.

Danny, I do not want to make work for you, but can you get a statistic on XY-wings? My impression is that both pincers are true about half the time.

Best wishes,

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Thu Jul 09, 2009 2:36 am    Post subject: Reply with quote

keith wrote:
Quote:
It's possible that they both may be true, but that's seldom the case.

Really? I thought it is quite common.

I think the simplest case of "one, or both, is true" is the pincers of an XY-wing.

Danny, I do not want to make work for you, but can you get a statistic on XY-wings? My impression is that both pincers are true about half the time.

First off, I overstepped myself in my claim above. A limited personal experience doesn't make something true. My apologies!!!

In the case of XY-Wings, I was able to find support for Keith position. I generated 299 VH puzzles and my solver encountered 628 XY-Wings while solving them. Of those, 339 had both pincers true -- 54%.

I'm still thinking over how this might apply to other scenarios where an at least one condition exists.

Regards, Danny

BTW: I now have 299 more VH puzzles to post. Very Happy

FWIW: I've spent time recently reviewing at least one conditions in URs that, for me, often turn out like the following.

Code:
 *** UR 2x bivalue cells:   <37> [r12c38]   cand count =  4/2,3,3,2

 [r3c3]=5 is true, and [r2c8]=6 is false
 +-----------------------------------------------------+
 |  2    567 *37+5 |  8    4    9    |  36  *37   1    |
 |  1    9   *37   |  26   27   5    |  8   *37+6 4    |
 |  67   8    4    |  36   37   1    |  9    2    5    |
 |-----------------+-----------------+-----------------|
 |  8    15   25   |  4    23   6    |  135  9    7    |
 |  4    157  257  |  23   9    8    |  135  35   6    |
 |  9    3    6    |  15   15   7    |  4    8    2    |
 |-----------------+-----------------+-----------------|
 |  67   4    1    |  57   8    3    |  2    56   9    |
 |  3    67   9    |  157  15   2    |  56   4    8    |
 |  5    2    8    |  9    6    4    |  7    1    3    |
 +-----------------------------------------------------+
 # 37 eliminations remain

Yes, I know there's a UR Type 1 present as well.
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Thu Jul 09, 2009 4:28 am    Post subject: Reply with quote

Danny,

Thank you! I don't know about UR's, and I really don't know about your puzzles.*

My comment was based on this site's many one-step XY-wing puzzles, and also my experience with Brain Bashers daily Super Hards.

But, if the logic says, "one", "or both" is true, I would start by presuming that each occurs about half the time.

Not that any of this helps to solve actual puzzles!

Keith

* What I mean is, there are probably no "real" statistics on Sudoku puzzles. All of the published ones are filtered for difficulty, are filtered to have a symmetric or almost-symmetric pattern in the initial clues, ...
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