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At an impasse (again)

 
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Wed Feb 22, 2006 10:09 pm    Post subject: At an impasse (again) Reply with quote

This was the "Tough" puzzle from sudoku.com.au for February 13.

As published:

Code:
|--6|3--|---|
|-5-|4--|-9-|
|7--|---|--2|
-------------
|6--|-5-|8--|
|--3|---|2--|
|--1|-8-|--6|
-------------
|2--|---|--8|
|-9-|--7|-5-|
|---|--1|6--|
-------------


And here's where I'm stuck. I've used two forcing chains, but they obviously didn't open things up. If someone can give me a hint on the next step...

Code:
49   1    6    3    279  259  457  8    457
38   5    2    4    67   68   137  9    137
7    38   49   59   1    589  345  6    2
6    247  479  127  5    2349 8    1347 179
4589 478  3    167  469  469  2    147  1579
459  247  1    27   8    2349 579  347  6
2    6    47   59   3    459  179  17   8
1    9    8    26   26   7    34   5    34
34   347  5    8    49   1    6    2    79
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Thu Feb 23, 2006 12:27 am    Post subject: Here's an idea ... Reply with quote

Hi, Marty! This is an interesting puzzle.
Code:
 49     1     6      3     279   259    457     8     457
 38     5     2      4     67    68     137     9     137
  7    38    49     59      1    589    345     6      2
  6    247  47+9    127     5   2349     8    1347    179
 4589  478    3     167    469   469     2     147   1579
 459   247    1     27      8   2349    579    347     6
  2     6    47-    59      3    459    179    17      8
  1     9     8     26     26     7     34      5     34
 34    347+   5      8     49     1      6      2     7-9

Observe that there are only two ways to fit a "7" in row 9 and in column 3, and that there are also only two ways to fit a "7" in the bottom left 3x3 box (see the "+" and "-" signs in the grid above). Clearly we either have r4c3 = 7, or else r9c9 = 7. Either way there cannot be a "7" at r4c9.

Now, with the possibilities at r4c9 reduced to {1, 9} we can see an XY-Wing pattern in r9c9. r7c8, and r4c9. If r9c9 = 7 then r7c8 = 1. And if r9c9 = 9 then r4c9 = 1. So there can't be a "1" at either r4c8 or r5c8, leaving r7c8 as the only possible spot to place a "1" in column 8.

That's not enough to solve the puzzle, but it does make a bit of progress. dcb Smile
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Thu Feb 23, 2006 1:21 am    Post subject: ... and another Reply with quote

A slightly different approach to the next two steps is:

(a) eliminate 5 from r2c1 using the finned X-Wing in rows 1 and 5 (which leaves possibles as (79)) and
(b) use the XY-Wing pivoted on r2c1 with pincers in cells r6c1 and r2c8 to eliminate 4 from r6c8.
There is then only one place for 4 in row 6 and the rest is routine.

Steve
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Thu Feb 23, 2006 7:27 pm    Post subject: Re: Here's an idea ... Reply with quote

David Bryant wrote:
Hi, Marty! This is an interesting puzzle.
Code:
 49     1     6      3     279   259    457     8     457
 38     5     2      4     67    68     137     9     137
  7    38    49     59      1    589    345     6      2
  6    247  47+9    127     5   2349     8    1347    179
 4589  478    3     167    469   469     2     147   1579
 459   247    1     27      8   2349    579    347     6
  2     6    47-    59      3    459    179    17      8
  1     9     8     26     26     7     34      5     34
 34    347+   5      8     49     1      6      2     7-9

Observe that there are only two ways to fit a "7" in row 9 and in column 3, and that there are also only two ways to fit a "7" in the bottom left 3x3 box (see the "+" and "-" signs in the grid above). Clearly we either have r4c3 = 7, or else r9c9 = 7. Either way there cannot be a "7" at r4c9.

Now, with the possibilities at r4c9 reduced to {1, 9} we can see an XY-Wing pattern in r9c9. r7c8, and r4c9. If r9c9 = 7 then r7c8 = 1. And if r9c9 = 9 then r4c9 = 1. So there can't be a "1" at either r4c8 or r5c8, leaving r7c8 as the only possible spot to place a "1" in column 8.

That's not enough to solve the puzzle, but it does make a bit of progress. dcb Smile

David, I follow your logic in the first paragraph which eliminates "7" from r4c9. However, spotting something like that appears to be beyond my powers of observation at this time. I also follow how you placed a "1" in r7c8, although you didn't say specifically, I believe the XY-Wing eliminates the "1" from r4c8.

I agree, it's not enough to solve the puzzle, but I'm more interested in learning a new technique, if indeed, I learned it. With the elimination of the "1" in column 8 of box 6, the remaining triple in column 8 allows removal of the other candidate "7" in that box, but that hasn't helped so far.

As always, thanks for the help.
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Thu Feb 23, 2006 7:33 pm    Post subject: Re: ... and another Reply with quote

Steve R wrote:
A slightly different approach to the next two steps is:

(a) eliminate 5 from r2c1 using the finned X-Wing in rows 1 and 5 (which leaves possibles as (79)) and
(b) use the XY-Wing pivoted on r2c1 with pincers in cells r6c1 and r2c8 to eliminate 4 from r6c8.
There is then only one place for 4 in row 6 and the rest is routine.

Steve

Steve, I don't understand. There was no "5" in r2c1 and the XY-Wing pivoted in r2c1 doesn't mesh with r6c1 and r2c8. ???

Is there a difference between a "finned X-Wing" and an "X-Wing"?

Thanks for taking the time to respond.
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Thu Feb 23, 2006 10:22 pm    Post subject: Solved!! Reply with quote

Three hours after my previous reply, it was solved with a couple of forcing chains. Thanks again for the little push.
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Fri Feb 24, 2006 1:47 am    Post subject: The real alternative Reply with quote

Sorry Marty: of course you didn’t understand it.

The story is this. Studying the puzzle and drafting my note took some time. When I came to post it, David had already answered so I scrubbed my effort. Then I thought it might be worth sending anyway, retrieved it, amended it slightly and sent it. The flaw in the procedure was retrieving the wrong note from the recycle bin. What I thought I sent was:

A slightly different approach to the next two steps is:

(a) eliminate 7 from r4c9 using the finned X-Wing in columns 3 and 8 (which leaves possibles as (19)) and
(b) use the XY-Wing pivoted on r9c9 with pincers in cells r7c8 and r4c9 to eliminate 1 from r45c8.
There is then only one place for 1 in column 8.

As for fins, an illustration may be of interest. Start with a normal X-Wing based on columns:

… A … C …
…………….
…………….
… B … D …

A and B are the only cells which may admit an entry, X, in the left-hand column; similarly for cells C and D in the right-hand column.

Now imagine cell C replaced by boxC, the box which contains cell C. Thus the left hand column still has only two places for X and they still line up with the column to the right. It is just that one of the “places” for X in the right hand column is three cells lined up in a box rather than a single cell.

Logic:
(1) Either cell A or cell B contains X.
(2) If cell A contains X, X cannot be placed in the intersection of row AC and boxC.
(3) If cell B contains X, then D cannot and the X in boxC lies in column CD.
(4) Whichever of (2) and (3) applies, X can be excluded from two cells in boxC. These cells comprise the intersection of row AC and boxC excluding cell C itself.

The pattern is less useful than an X-Wing but, if you look for X-Wings at all, you will spot the finned version just as easily. Incidentally, any m x m fish can have a fin (where m = 2 is an X-Wing, m = 3 a swordfish, etc). By coincidence a finned swordfish can be applied later in the same puzzle.

Steve
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Fri Feb 24, 2006 6:29 pm    Post subject: Reply with quote

Thanks Steve, for clearing that up. I've printed out your reply, as I did David's, and will pore over it to see what I can learn.
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