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arkietech
Joined: 31 Jul 2008
Posts: 1834
Location: Northwest Arkansas USA
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 Posted: Fri Nov 14, 2008 2:56 pm Post subject: Pattern entry from GSF |
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| Code: |
+-------+-------+-------+
| 3 . . | . . . | . . 7 |
| . 2 . | . 1 . | . 9 . |
| . . 5 | 6 . 7 | 8 . . |
+-------+-------+-------+
| . . 4 | 8 . 5 | 1 . . |
| . . . | . . . | . . . |
| . . 1 | 9 . 4 | 5 . . |
+-------+-------+-------+
| . . 2 | 3 . 9 | 4 . . |
| . 4 . | . 8 . | . 6 . |
| 9 . . | . . . | . . 8 |
+-------+-------+-------+
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Play this puzzle online at the Daily Sudoku site
Once you start you are finished.
| Quote: | | I think it is another one of those m or w wings |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Fri Nov 14, 2008 4:23 pm Post subject: |
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With PMs it is easier to see...
(and actually it's the first move anyway)
so here goes.
BEFORE basics (because there are none  )
| Code: |
+--------------------------+--------------------------+--------------------------+
| 3 1689 689 | 245 2459 28 | 26 1245 7 |
| 4678 2 678 | 45 1 38 | 36 9 3456 |
| 14 19 5 | 6 2349 7 | 8 1234 1234 |
+--------------------------+--------------------------+--------------------------+
| 267 3679 4 | 8 2367 5 | 1 237 2369 |
| 25678 356789 36789 | 127 2367 1236 | 23679 23478 23469 |
| 2678 3678 1 | 9 2367 4 | 5 2378 236 |
+--------------------------+--------------------------+--------------------------+
| 15678 15678 2 | 3 567 9 | 4 157 15 |
| 157 4 37 | 1257 8 12 | 2379 6 12359 |
| 9 13567 367 | 12457 24567 126 | 237 12357 8 |
+--------------------------+--------------------------+--------------------------+
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look at r12c67.
We've got an xy-loop here.
Outside the four cells:
- no 2 in row 1
- no 8 in col 6 or box 2 (there aren't any)
- no 3 in row 2
- no 6 in col 7 or in box 3
This solves the puzzle
edit: removed false assertion "(none either)" from: no 6 in col 7
edit again to include "and box ..." 6 in r2c9 sees both r1c7 and r2c7 and can be eliminated.
Last edited by nataraj on Fri Nov 14, 2008 5:41 pm; edited 2 times in total |
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arkietech
Joined: 31 Jul 2008
Posts: 1834
Location: Northwest Arkansas USA
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| Posted: Fri Nov 14, 2008 4:31 pm Post subject: |
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| nataraj wrote: | With PMs it is easier to see...
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This is what I found | Code: | *-----------------------------------------------------------------------------*
| 3 1689 689 | 45-2 459-2 *28 |*26 145-2 7 |
| 4678 2 678 | 45 1 *38 |*36 9 3456 |
| 14 19 5 | 6 2349 7 | 8 1234 1234 |
|-------------------------+-------------------------+-------------------------|
| 267 3679 4 | 8 2367 5 | 1 237 2369 |
| 25678 356789 36789 | 127 2367 1236 | 23679 23478 23469 |
| 2678 3678 1 | 9 2367 4 | 5 2378 236 |
|-------------------------+-------------------------+-------------------------|
| 15678 15678 2 | 3 567 9 | 4 157 15 |
| 157 4 37 | 1257 8 12 | 2379 6 12359 |
| 9 13567 367 | 12457 24567 126 | 237 12357 8 |
*-----------------------------------------------------------------------------*
either r1c6 or r1c7 = 2 what do you call it? |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Fri Nov 14, 2008 5:40 pm Post subject: |
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| Quote: | | what do you call it |
dan,
what you found is a short xy-chain 28-83-36-62, a.k.a "extended" xy-wing (because the 28 and the 26 cell are not linked by a single 68 cell, but a two-cell combination that serves the same purpose: 63-38)
Looking a little closer, the same four cells form another extended xy-wing:
82-26-63-38: at least one of r1c6 and r2c6 must be 8.
There is another extended xy-wing. You guessed:
62-28-83-36: at least one of r12c7 must be 6
And one more:
36-62-28-83. At least one of r2c67 must be 3.
This very nice pattern (overlapping xy-chains that work round robin) is called an xy-loop. |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Fri Nov 14, 2008 7:12 pm Post subject: |
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A less obvious/pretty solution is a continuous loop in cells a-b-c-dd-a for (4) & (5). This leads to Singles completing the puzzle.
| Code: | +--------------------------------------------------------------------------------+
| 3 1689 689 | d245 d2459 28 | 26 a45-12 7 |
| 4678 2 678 | c45 1 38 | 36 9 b3456 |
| 14 19 5 | 6 239-4 7 | 8 1234 1234 |
|--------------------------+--------------------------+--------------------------|
| 267 3679 4 | 8 2367 5 | 1 237 2369 |
| 25678 356789 36789 | 127 2367 1236 | 23679 23478 23469 |
| 2678 3678 1 | 9 2367 4 | 5 2378 236 |
|--------------------------+--------------------------+--------------------------|
| 15678 15678 2 | 3 567 9 | 4 157 15 |
| 157 4 37 | 1257 8 12 | 2379 6 12359 |
| 9 13567 367 | 12457 24567 126 | 237 12357 8 |
+--------------------------------------------------------------------------------+
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Fri Nov 14, 2008 8:43 pm Post subject: |
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Dan,
its called a loop because the chain keeps going around, never ends.
in the loop 28-38-36-26, the bi-value cell acts as the strong inference and the links between the cells (the common number) is the weak inference. you can cut the loop in any of the weak inferences (between cells) and still have a valid loop.
if you are still curious about the implications of such loops and chains, here is a very good tutorial.
http://www.sudoku.com/boards/viewtopic.php?t=3865
sadly this tutorial doesn't give a x-wing example.
a x-wing uses this same logic. a x-wing is just a special case of a loop on one candidate. some might call this a cycle. other names for one candidate loops, swordfish that use 6 cells, and jellyfish that use 8 cells. obviously these loops don't need to be in perfect rectangular patterns. the most important thing to remember is that the candidates with the weak inferences are what make the eliminations around the loop. the weakly linked candidates can't both be true, but the loop says one is true, thus eliminating any other candidate they see. this also means that if the weakly linked candidates share a cell, all other candidates in the cell are gone which is one of those rules I had trouble grasping and remembering.
another example the tutorial doesn't address is the simplest of them all, the heralded "naked pair" which saves us from those "hard" puzzles all the time.
(5=6) - (6=5)
as you can see the 6's are weakly linked, but what you can't see is that the 5's are too. this creates the loop and it says that either one of the 5's or either one of the 6's is true. any other 5's and 6's they see is eliminated from the row, col, or box.
hidden pair: perfect example for a loop that include cells that contain more than two candidates.
for this I am going to use nataraj's "not vh" puzzle
| Code: |
+------------------+--------------------+-----------------+
| 239 2359 8 | 1 49 6 | 39 49 7 |
| 79 6 4 | 5789 3 5789 | 189 2 189 |
| 2379 12379 129 | 24789 4789 2789 | 5 4689 13489 |
+------------------+--------------------+-----------------+
| 2389 2359 259 | 5789 1789 135789 | 4 5789 6 |
| 689 4 569 | 56789 2 15789 | 189 3 1589 |
| 1 2359 7 |#45689#4689 3589 | 89 589 2589 |
+------------------+--------------------+-----------------+
| 24679 1279 3 | 26789 16789 12789 | 689 45689 4589 |
| 4679 8 69 | 679 5 79 | 2 1 349 |
| 5 129 1269 | 3 1689 4 | 7 689 89 |
+------------------+--------------------+-----------------+
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Play this puzzle online at the Daily Sudoku site
notice in r6c45 the marked cells each contain the candidates 4 and 6
making this loop (6)r6c4 = (6-4)r6c5 = (4)r6c4; r6c45 <>5,8,9
notice how the weak link happens inside the cell. this chain is a loop because the 4 in r6c4 is weakly linked to the 6 in r6c4.
this means that either the 4 or the 6 is true in r6c4 and either the 4 or the 6 is true in r6c5. and since each individual candidate is strongly linked, if one is true forcing the other false, then the false candidate must be true in the other cell. whew !!
this eliminates all other candidates in r6c45
simply put, these cells can only contain {4,6}. |
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arkietech
Joined: 31 Jul 2008
Posts: 1834
Location: Northwest Arkansas USA
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| Posted: Fri Nov 14, 2008 10:07 pm Post subject: |
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| storm_norm wrote: | its called a loop because the chain keeps going around, never ends.
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Thanks Norm, it has been a good day for me with new vistas opening in sudoku. These concepts are powerful in hard puzzles.  |
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Marty R.
Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA
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| Posted: Fri Nov 14, 2008 10:27 pm Post subject: |
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| Quote: | notice in r6c45 the marked cells each contain the candidates 4 and 6
making this loop (6)r6c4 = (6-4)r6c5 = (4)r6c4; r6c45 <>5,8,9
notice how the weak link happens inside the cell. this chain is a loop because the 4 in r6c4 is weakly linked to the 6 in r6c4.
this means that either the 4 or the 6 is true in r6c4 and either the 4 or the 6 is true in r6c5. and since each individual candidate is strongly linked, if one is true forcing the other false, then the false candidate must be true in the other cell. whew !!
this eliminates all other candidates in r6c45
simply put, these cells can only contain {4,6}. |
Norm, isn't this a little complicated for what is a simple quint in row 6? |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Fri Nov 14, 2008 10:43 pm Post subject: |
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| Quote: | | Norm, isn't this a little complicated for what is a simple quint in row 6? |
Marty,
in nataraj's puzzle, if I only view the puzzle with all possible 4's I can see that a 4 can only go in two places in row 6.
AND
if I only view the puzzle with all possible 6's I see that 6 can only be placed in two cells in row 6.
in fact the two cells match. r6c45.
this tells me that in row 6, the pair {4,6} are the only two candidates that fit into r6c45.
the AIC explanation (if I got it right) was to link the similar logic involved when viewing it as a loop. and you are right, its probably a little complicated for a simple technique.
Last edited by storm_norm on Fri Nov 14, 2008 11:09 pm; edited 1 time in total |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Fri Nov 14, 2008 11:08 pm Post subject: |
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Where's Asellus when you need him?
[Withdrawn: I was incorrect in how I presented continuous and discontinuous loops.]
Last edited by daj95376 on Mon Nov 17, 2008 5:15 am; edited 2 times in total |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Fri Nov 14, 2008 11:19 pm Post subject: |
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| daj95376 wrote: | | Where's Asellus when you need him? This thread is in serious need of an explanation of the distinction between a continuous loop and a discontinuous loop. |
i'd love to see your definitions. |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Fri Nov 14, 2008 11:28 pm Post subject: |
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| storm_norm wrote: | | daj95376 wrote: | | Where's Asellus when you need him? This thread is in serious need of an explanation of the distinction between a continuous loop and a discontinuous loop. |
i'd love to see your definitions. |
I was expanding my message as you were entering yours. Look UP. |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Sat Nov 15, 2008 12:14 am Post subject: |
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| Quote: | A discontinuous loop stops when it returns to the originating cell. Both are called loops because they return to the originating cell.
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in AIC's. the chain always starts with a strong inference and ends with a strong inference. if the chain ends in the same "house" then the two ends must be weakly linked in order to form the loop.
it doesn't matter whether they end in same cell or not. if the chain ends in the same row, box or column, then the chain needs to end on the same candidate that it started. this is what creates the weak inference between the start of the chain to the end.
if the chain ends in the same cell, then the loop by definition is complete because the cell itself provides the weak inference, since both ends are true.
just examples, not real chains.
(8)r6c4 = (8)r6c9 - (8=1)r1c9 - (1=2)r1c5 - (2=8)r1c4
this chain ends and starts in col 4, the 8's are then weakly linked (even if they are conjugate) completing the loop.
(8)r6c4 = (8-1)r6c9 = (1)r1c9 - (1)r1c4 = (1)r6c4
this chain ends in the same cell, the 1 and the 8 are weakly linked because both can't be true, thus completing the loop.
lets see if I can give discontinous nice loops a shot.
in AIC notation, A = - = - =B (this ending cell also contains A)
the beginning cell containing A can see the ending cell containing B.
but the ending cell contains A also.
in the ending cell, A-B, A is weakly linked to B, but at the same time, B is not weakly linked to A in the beginning cell. ( if it were it would be a loop.)
in other words, you can't prove that both A(beginning cell) and B(ending cell) can't both be true.
there is the discontinuity. the weak link couldn't be established. but the chain does say that either A is true or B is true and A can be eliminated from the ending cell. |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Sat Nov 15, 2008 12:27 am Post subject: |
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| Quote: | | I'm out of my league here, but I believe that a continuous loop always eliminates all but two candidates in the starting cell. For an XY-loop, this isn't apparent. In the continuous loop that I presented, it is apparent. |
in AIC, in a nice loop, any weakly linked candidates can't be both true, but the loop says that one can be. so the weakly linked candidates eliminate any other candidates that they see.
any other way to state this? |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sat Nov 15, 2008 3:50 am Post subject: |
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[Withdrawn: I was incorrect in how I presented continuous and discontinuous loops.]
Last edited by daj95376 on Mon Nov 17, 2008 5:16 am; edited 1 time in total |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sat Nov 15, 2008 12:19 pm Post subject: |
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[Withdrawn: I was incorrect in how I presented continuous and discontinuous loops.]
Last edited by daj95376 on Mon Nov 17, 2008 5:17 am; edited 1 time in total |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Sat Nov 15, 2008 8:40 pm Post subject: |
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| Quote: | | [r8c6]=7=[r7c4]=1=[r7c3]-1-[r6c3]-9-[r56c2]=9=[r8c2]=7=[r8c6] => [r8c6]<>18 |
this is how that chain would look in Eureka!.
(7)r8c6 = (7)r7c4 - (7=1)r7c3 - (1=9)r6c3 - (9)r56c2 = (9)r8c2; r8c2 <> 7
from my understanding of AIC's,
there exists a 7 in r8c2.
the AIC chain above says that either the 7 in r8c6 is true or the 9 in r8c2 is true. in either case, the 7 in r8c2 cannont exist.
your NL chain says this
(7)r8c6 = (7)r7c4 - (7=1)r7c3 - (1=9)r6c3 - (9)r56c2 = (9)r8c2 - (7)r8c2 = (7)r8c6
this chain says that either the 7 at the beginning is true, or the 7 at the ending is true.  the 7 in question is the same 7. this is not a contradiction. in fact, this also proves that the 7 in r8c2 cannot exist.
now take the last link off
(7)r8c6 = (7)r7c4 - (7=1)r7c3 - (1=9)r6c3 - (9)r56c2 = (9)r8c2 - (7)r8c2 ????
ahh, now we see the discontinuity. the 7 in r8c2 cannot be weakly linked to the beginning in order to form the loop because that would snap the alternating strong/weak inference cycle.
Danny, i am not disagreeing that your NL chain would make r8c6 <> 18, I just wanted to provide the AIC alternative. i would have to study NL notation a lot more to see it. |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sat Nov 15, 2008 10:03 pm Post subject: |
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[Withdrawn: I was incorrect in how I presented continuous and discontinuous loops.]
Last edited by daj95376 on Mon Nov 17, 2008 5:18 am; edited 1 time in total |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Sun Nov 16, 2008 6:33 am Post subject: |
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Danny,
yes, I see the contradiction. sorry for not being caught up on NL notation and all of its implications. I guess my confusion is why this means that r8c6 is not 1 and 8.
| Quote: | meant [r8c6]<>7 as well -- especially since you want your chain to be an AIC like mine. Combine this with your chain ending:
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remember that when the chain is formed, it means that either of the two ends is true, not both!
it doesn't mean, in a literal sense, that 7 is taken out of r8c6.
in fact, its stating that either the 7 in r8c6 exists or the other end of the chain exists.
think about your chain. if you remove the 7 from r8c6, and follow the chain to where 9 is forced into r8c2, then all of the 7's in row 8 are gone. poof! this makes me scratch my head because hmm, which 7 is being forced back into r8c6 if none now exist? |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sun Nov 16, 2008 9:12 am Post subject: |
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[Withdrawn: I was incorrect in how I presented continuous and discontinuous loops.]
Last edited by daj95376 on Mon Nov 17, 2008 5:18 am; edited 1 time in total |
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