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ravel
Joined: 21 Apr 2006
Posts: 536
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 Posted: Mon Aug 04, 2008 10:40 am Post subject: Too hard ? Then remove a given ... |
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And old thread of mine, how a puzzle can become harder with an additional clue, was warmed up these days by TTHsieh, who posted a couple of new examples.
So here are some puzzles (with increasing difficulty), which become easier to solve, when you remove a given.
| Code: | +-------+-------+-------+
| . . . | . . 1 | . 2 3 |
| . . . | . 4 . | . 5 6 |
| . . . | . . . | 7 . 8 |
+-------+-------+-------+
| 1 . . | . . 6 | 8 . . |
| 5 3 . | . 2 . | . 9 7 |
| . . 9 | 7 . . | . . 4 |
+-------+-------+-------+
| 6 . 3 | . . . | . . . |
| 4 2 . | . 5 . | . . . |
| . 8 . | 3 . . | . . . |
+-------+-------+-------+ TTHsieh
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To make it easier, remove 3 in r1c9
| Code: | +-------+-------+-------+
| 1 2 . | 3 . . | . . 4 |
| . . 5 | . . 4 | . 3 . |
| . . . | . 5 . | 2 . . |
+-------+-------+-------+
| . 6 . | . . 7 | . . 5 |
| 8 . . | . . . | . . 3 |
| 9 . . | 1 . . | . 2 . |
+-------+-------+-------+
| . . 4 | . 6 . | . . . |
| . 3 . | 9 . . | 7 . . |
| 6 . . | . . 2 | . 8 . |
+-------+-------+-------+ TTHsieh
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Remove 1 in r1c1
| Code: | +-------+-------+-------+
| 1 2 3 | . . . | . . . |
| . . . | . 4 . | 5 . . |
| . 5 . | . 2 6 | . 7 . |
+-------+-------+-------+
| . 8 . | 9 . . | . . 4 |
| . . 4 | . . . | 2 . . |
| 3 . . | . . 4 | . 6 . |
+-------+-------+-------+
| . 9 . | 6 1 . | . 2 . |
| . . 5 | . 8 . | . . . |
| . . . | . . . | 6 . 8 |
+-------+-------+-------+ TTHsieh
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Remove 2 in r1c2
| Code: | +-------+-------+-------+
| 1 2 . | 3 . . | . . 4 |
| . . 5 | . . 4 | . 3 . |
| . . . | . 6 . | 2 . . |
+-------+-------+-------+
| . 7 . | . . 8 | . . 9 |
| 6 . . | . . . | . . 3 |
| 9 . . | 1 . . | . 2 . |
+-------+-------+-------+
| . . 4 | . 7 . | . . . |
| . 3 . | 8 . . | 1 . . |
| 7 . . | . . 2 | . 5 . |
+-------+-------+-------+ TTHsieh
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Remove 1 in r1c1
| Code: | +-------+-------+-------+
| . . 1 | . . . | 2 . . |
| . . . | 3 . 4 | . . . |
| 5 . . | . 6 . | . . 7 |
+-------+-------+-------+
| . 6 . | 4 . 8 | . 2 . |
| . . 2 | 5 . . | 9 . . |
| . 3 . | 6 . 2 | . 8 . |
+-------+-------+-------+
| 7 . . | . 8 . | . . 1 |
| . . . | 1 . 6 | . . . |
| . . 9 | . . . | 3 . . |
+-------+-------+-------+ TTHsieh
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Remove 5 in r5c4 |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Mon Aug 04, 2008 5:26 pm Post subject: |
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CAUTION, WET FLOOR!!!!
all the following patterns are NOT deadly, because they contain a given !
I leave them in just the same, I think they illustrate a possible search pattern, if not an actual solution....
This is what I get when solving the first puzzle:
| Code: |
+--------------------------+--------------------------+--------------------------+
| 78 45 456 | 568 678 1 | 9 2 3 |
| 23789 79 278 | 289 4 2378 | 1 5 6 |
| 239 15 156 | 2569 36 2359 | 7 4 8 |
+--------------------------+--------------------------+--------------------------+
| 1 47 247 | 45 9 6 | 8 3 25 |
| 5 3 48 | 1 2 48 | 6 9 7 |
| 28 6 9 | 7 38 358 | 25 1 4 |
+--------------------------+--------------------------+--------------------------+
| 6 1579 3 | 2489 178 24789 | 245 78 1259 |
| 4 2 17 | 689 5 789 | 3 678 19 |
| 79 8 157 | 3 167 2479 | 245 67 125 |
+--------------------------+--------------------------+--------------------------+
|
Following a suggestion by Asellus (and ravel's hint r1c9, of course) I looked at the "almost DP" in r18c79: If r8c9 were 9, that would constitute the DP already. So r8c9 must be 1.
The interesting question, for me, is: why is this pattern harder to find than the usual 39 39 39 139 rectangle we get after removing the given 3 in r1c9? I guess because we are not trained to look for such patterns, maybe? Or - more probably - because the repetition really attracts our attention and singles, like in the above grid, are too plain ...
Very interesting thread, ravel!
----
edit 2009 GMT+2: coming back to this position with my new way to look at these positions, I realize there is at least another almost-DP: 23 in r14c89 which sets r4c9=5
edit again 2012: and what about that almost DP 15 in r26c78? .... I am definitely starting to like this a lot
edit again 2015: another DP 38 r48c78 makes 67 in col 8 a naked pair...
another possible DP 15 r45c14 - found 2018
Last edited by nataraj on Mon Aug 04, 2008 6:44 pm; edited 6 times in total |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Mon Aug 04, 2008 5:38 pm Post subject: |
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Second puzzle, same story.:
Removing the 1 in r1c1, there is a UR type 4 (17) and 1 can be removed from r12c5.
But the almost DP is there even in the position we get from the original puzzle:
| Code: |
+--------------------------+--------------------------+--------------------------+
| 1 2 689 | 3 7 689 | 5689 569 4 |
| 7 89 5 | 268 1289 4 | 1689 3 1689 |
| 34 489 3689 | 68 5 1689 | 2 1679 16789 |
+--------------------------+--------------------------+--------------------------+
| 34 6 123 | 248 23489 7 | 1489 149 5 |
| 8 1457 127 | 2456 249 569 | 1469 14679 3 |
| 9 457 37 | 1 348 3568 | 468 2 678 |
+--------------------------+--------------------------+--------------------------+
| 25 189 4 | 7 6 1358 | 1359 159 129 |
| 25 3 18 | 9 148 158 | 7 1456 126 |
| 6 179 179 | 45 34 2 | 345 8 19 |
+--------------------------+--------------------------+--------------------------+
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If r2c5 were 1, that would be a DP, so we can remove 1 from r2c5 (same elimination as the one resulting from the type 4)
So, the question again (but this time harder to answer): is it easier to see a type 4 UR or to see the pattern:
hm .... (intrigued)  |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Mon Aug 04, 2008 5:56 pm Post subject: |
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Without removing 2 from r1c2:
| Code: |
+--------------------------+--------------------------+--------------------------+
| 1 2 3 | 578 579 5789 | 89 4 6 |
| 789 67 689 | 13 4 1389 | 5 1389 2 |
| 4 5 89 | 138 2 6 | 1389 7 139 |
+--------------------------+--------------------------+--------------------------+
| 57 8 267 | 9 3567 12357 | 137 135 4 |
| 579 167 4 | 13578 3567 13578 | 2 13589 13579 |
| 3 17 279 | 12578 57 4 | 1789 6 1579 |
+--------------------------+--------------------------+--------------------------+
| 78 9 78 | 6 1 35 | 4 2 35 |
| 6 34 5 | 24 8 239 | 1379 139 1379 |
| 2 34 1 | 457 3579 3579 | 6 359 8 |
+--------------------------+--------------------------+--------------------------+
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Can you find the almost-DP here?
Hint: The DP is 26 in r12c29 and in order to avoid it r2c2 must be 7. |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Mon Aug 04, 2008 6:05 pm Post subject: |
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Forth puzzle (again without removal) leads to:
| Code: |
+--------------------------+--------------------------+--------------------------+
| 1 2 679 | 3 8 579 | 5679 679 4 |
| 8 69 5 | 279 129 4 | 679 3 167 |
| 34 49 379 | 579 6 1579 | 2 789 1578 |
+--------------------------+--------------------------+--------------------------+
| 34 7 123 | 2456 2345 8 | 456 146 9 |
| 6 1458 128 | 24579 2459 79 | 4578 1478 3 |
| 9 458 38 | 1 345 367 | 45678 2 5678 |
+--------------------------+--------------------------+--------------------------+
| 25 1689 4 | 569 7 13 | 3689 689 268 |
| 25 3 69 | 8 459 569 | 1 4679 267 |
| 7 1689 1689 | 469 13 2 | 34689 5 68 |
+--------------------------+--------------------------+--------------------------+
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Maybe I'm getting better at this, maybe it is because the DP is in the top left. Must be much harder to see when buried inside the puzzle.
Hint: DP 18 in r12c15. |
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ravel
Joined: 21 Apr 2006
Posts: 536
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| Posted: Mon Aug 04, 2008 6:25 pm Post subject: |
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Just for clarification:
If you try to spot UR's with solved numbers, you have to ensure, that none of these numbers is a given. Otherwise the pattern is not deadly, because in a solution with the pattern you cannot xx the numbers to get a second one.
In these puzzles the mentioned numbers can be removed, because they are redundant to give a unique solution. But if this is not ensured to be an additional property, you would have to solve it without the uniqueness elimination. |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Mon Aug 04, 2008 6:31 pm Post subject: |
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Now that I've finished scanning for possible URs in ravel's first puzzle and having found 5 (five!) edit:**** FOOL's GOLD !!! **** potential DP eliminations in an otherwise "unsolvable" puzzle, I am really amazed at the effectiveness of this new method.
All I did was scan line by line and look at all the singles in turn.
Like, in the first row, there are 4 singles: 1,9,2,3. For every single I looked at the columns and when I found another one out of the four, that would be a possible candidate.
This turned up the "original" 39 DP, but also the 23 (not D)P.
Row 2 yields the 15 (not D)P, row 4 the 38 (not D)P.
And so on ...
Not a bad start into the week for a Monday night,
Thanks, ravel !!
Last edited by nataraj on Mon Aug 04, 2008 6:48 pm; edited 2 times in total |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Mon Aug 04, 2008 6:32 pm Post subject: |
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| ravel wrote: | Just for clarification:
If you try to spot UR's with solved numbers, you have to ensure, that none of these numbers is a given. Otherwise the pattern is not deadly, because in a solution with the pattern you cannot xx the numbers to get a second one.
In these puzzles the mentioned numbers can be removed, because they are redundant to give a unique solution. But if this is not ensured to be an additional property, you would have to solve it without the uniqueness elimination. |
OOPS ... in my excitement I forgot ...
So these are genuine examples of puzzles actually getting easier by removing a (redundant) given, not just make the crucial move easier to spot ...
Hm ... no help in solving actual sudokus then ... I was soooo hoping for a new "silver bullet" 
Still, I'm going to keep a lookout for these hidden DPs. On paper it should be much easier to avoid the trap, since the givens are printed and would stand out quite conspicuously... Even fewer singles to check, then. |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Mon Aug 04, 2008 10:25 pm Post subject: |
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| nataraj wrote: | | Following a suggestion by Asellus |
I hope ravel has made it clear that in my previous suggestion about DPs I did not mean to imply that you can use DP logic when one or more of the digits in what would otherwise be a DP is fixed as a given. It worked in these puzzles only because they were specially selected. But the approach is not valid.
Our new friend the Finned XY Wing helped quite a bit in the "harder" version of the first puzzle. |
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wapati
Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.
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| Posted: Mon Aug 04, 2008 11:31 pm Post subject: |
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Puzzle 3 is as easy with and without.
One finned x-wing on 8s removing 8 from r3c4, and from r2c4 if you have it still there.
I think the original thread was about adding a clue to make a puzzle harder.
All but #3 fit that, they become testy.  |
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tlanglet
Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills
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| Posted: Tue Aug 05, 2008 2:53 am Post subject: |
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| Asellus wrote: |
Our new friend the Finned XY Wing helped quite a bit in the "harder" version of the first puzzle. |
I attempted to find a fruitful finned xy-wing based on Asellus's comment but was not successful. I did find three opportunities, but I could not prove valid eliminations. It is obvious that I do not have this new technique under control.
Would someone help me once again? 
Thanks for your assistance.
Ted |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Tue Aug 05, 2008 5:22 am Post subject: |
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| Asellus wrote: |
I hope ravel has made it clear that in my previous suggestion about DPs I did not mean to imply that you can use DP logic when one or more of the digits in what would otherwise be a DP is fixed as a given. |
That was very clear, Asellus, and this is another example of a perfectly good tool used in a wrong way (by me). I was referring to your post in the other thread about not adding in a candidate (in order to make a UR visible) but instead to recognize the deadly pattern even when some cells are already singles.
I guess that is how we learn: by making mistakes and having good teachers to point them out and correct them |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Tue Aug 05, 2008 5:35 am Post subject: |
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| Quote: | | I think the original thread was about adding a clue to make a puzzle harder. |
Add a clue to puzzle A, making a harder puzzle, B.
Remove a clue from puzzle B, making an easier puzzle A.
Are these not the same thing?
I think Nararaj' reasoning is correct, but he is using extra information. This is that puzzle A has a unique solution. Even then, the logic is not precise unless you also know which clue was added. (To be more precise, the information is that A and B have the same, unique, solution.)
Suppose puzzle A has this fragment:
The UR results in
Now, suppose puzzle B has this fragment:
where the upper left cell is a given initial clue. If I know it came from A, I can write
If I do not know it came from A, there is no reason to exclude
which is NOT a DP, since the top left value is a given:
can not be a solution.
Anyway, I will be away for three weeks, with very limited access to e-mail and the net. (Such are the facts in rural Africa.) Have fun, be good without me. (Don't give Marty too much work.)
Keith |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Tue Aug 05, 2008 5:49 am Post subject: |
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| Quote: | | Our new friend the Finned XY Wing helped quite a bit in the "harder" version of the first puzzle. |
Asellus,
Could you please post the details? (I have not had the time to work these puzzles.) I am very pleased that we have gone from the idea of a finned XY-wing to a good understanding in only two weeks, but I think we still need to look at the details of examples as we find them.
Keith |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Tue Aug 05, 2008 9:57 am Post subject: |
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| keith wrote: | | Could you please post the details? |
Happily. Here's the grid for puzzle 1 after basics:
| Code: | +------------------+------------------+----------------+
| 78 45 456 | 568 678 1 | 9 2 3 |
| 23789 79 278 | 289 4 23789 | 1 5 6 |
| 239 15 156 | 2569 36 2359 | 7 4 8 |
+------------------+------------------+----------------+
| 1 47 247 | 45 9 6 | 8 3 25 |
| 5 3 48 | 1 2 48 | 6 9 7 |
| 28 6 9 | 7 38 358 | 25 1 4 |
+------------------+------------------+----------------+
| 6 1579 3 | 2489 178 24789 | 245 78 1259 |
| 4 2 17 | 689 5 789 | 3 678 19 |
| 79 8 157 | 3 167 2479 | 245 67 1259 |
+------------------+------------------+----------------+ |
Two steps are needed to get to the grid nataraj posted:
[1] 179 XY Wing r89 removes 9 from r9c9
[2] Finned X-Wing r39 removes 9 from r2c6
(Can also be seen as a Kite or an ER elimination.)
Now, in nataraj's grid:
[3] Sue de Coq r1236c1|r2c2 removes 7 from r2c3
(I haven't checked... not sure if this step is required. But, it's interesting anyway.)
Now comes the Finned XY Wing...
[4] Potential 238 XY Wing in r3c16|r6c1 with Fin=9 in r3c1:
(9)r3c1=[XY Wing]-(3=6)r3c5-(6)r9c5=(6)r9c8
and
(9)r3c1-(9=7)r9c1-(7=6)r9c8; r9c8=6
For those not comfortable with the Eureka notation... If the <9> Fin is true, then a short XY Chain (via r9c1) determines that r9c8=6. If the XY Wing is true, then r3c5=6 and the strong link on <6> in r9 determines that r9c8=6. Either way, r9c8=6.
Two more XY Wings solve the puzzle:
[5] 458 XY Wing r1c24|r4c4 removes 4 from r4c2
[6] 238 XY Wing r3c1|r6c15 removes 3 from r3c5 |
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tlanglet
Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills
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| Posted: Tue Aug 05, 2008 2:54 pm Post subject: |
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Thanks for the details on the finned xy-wing Asellus.
I believe that my approach for the three potential finned xy-wings I attacked was correct, and that I simply did not find the <238> condition you described before I stopped in frustration.
Thanks again ..........
Ted |
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Marty R.
Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA
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| Posted: Tue Aug 05, 2008 4:30 pm Post subject: |
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| Quote: | | If the <9> Fin is true, then a short XY Chain (via r9c1) determines that r9c8=6. If the XY Wing is true, then r3c5=6 and the strong link on <6> in r9 determines that r9c8=6. Either way, r9c8=6. |
| Code: |
+----------+----------+------------+
| 6 3 1 | 58 7 9 | 4 58 2 |
| 79 4 789 | 58 1 2 | 358 36 678 |
| 5 2 78 | 3 6 4 | 9 1 78 |
+----------+----------+------------+
| 2 6 4 | 7 3 5 | 18 9 18 |
| 8 7 5 | 6 9 1 | 2 4 3 |
| 1 9 3 | 4 2 8 | 6 7 5 |
+----------+----------+------------+
| 79 8 6 | 2 4 37 | 157 35 19 |
| 4 5 79 | 1 8 367 | 37 2 69 |
| 3 1 2 | 9 5 67 | 78 68 4 |
+----------+----------+------------+
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Play this puzzle online at the Daily Sudoku site
Note the Finned XY-Wing, with the fin <7> in the pivot cell, r8c6. I've discarded the notes I had, but I think that solves a lot of cells and the whole puzzle.
Note the <78> UR in boxes 13. One of the roof cells must be <6> or <9>, forcing r3c3=8.
Can we have a Finned naked triple? Look at r7c678. R7c7=1 or else we have a <357> triple, which, if I remember correctly, also forces a few cells.
I view all three cases as the same, i.e., something is spotted which directs the player to a particular forcing chain. In other words, these techniques border on trial-and-error, the way I see things.
A few months ago I asked Keith about the T&E aspects of a solution he found using a UR like the aforementioned one. His answer, a good one, as one would expect, was that there was nothing wrong with spotting a pattern and determining its implications.
Nevertheless, I was never 100% comfortable with the idea that these contained no T&E element, even though I use these techniques. But how different is a regular forcing chain, such as looking at column 7 and realizing that r2 or r8 must be =3?
I'm very open to someone punching wide holes in my theory. Then I'd feel better about these things.
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Tue Aug 05, 2008 5:34 pm Post subject: |
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| Marty R. wrote: | But how different is a regular forcing chain, such as looking at column 7 and realizing that r2 or r8 must be =3?
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I think the 3s in col 7 are as far from a forcing chain as one can get. No chain at all, simply a (strong) link. No implication yet, only an observation: there are exactly two 3s - one of the two cells must be 3.
Look at any single in the grid: r1c1 is 6. That means no other cell in row 1 can be 6. A forcing chain? Not yet.
I'll give an example of a forcing chain...
I randomly select the 79 in r2c1, and randomly assume r2c1=9. Let's see where it leads (don't try this at home, folks. Chains like this have a nasty habit of ending up in nothing at all)
if r2c1=9 then there is a naked pair 78 in col 3 then r8c3=9 then r8c9=6 (a) then naked pair 78 in col 9 then r4c9=1(b).
But if (a)r8c9=6 then r9c8=8(c) then r9c7=7 then r8c7=3 (d) AND
if (c)r9c8=8 then r1c8=5
Together with (d) that means r2c7=8 and r4c7=1, which is a contradiction to (b).
Taking it all together, we have proven that r2c1 cannot be 9. Logically sound, yes. But elegant? And I really really DID choose r2c1 at random...
Where does pattern search end and forcing begin?
Good question.
Next question, please ...
Is there even a difference?
Hard to say.
On a totally unrelated topic : is there such a thing as "quality"? (Robert M. Pirsig comes to mind )
Some say "define quality". Others just know it exists. Still others claim they can see whether an object has quality or not...
What about this sentence then?
| Quote: | | If the <9> Fin is true, then a short XY Chain (via r9c1) determines that r9c8=6. If the XY Wing is true, then r3c5=6 and the strong link on <6> in r9 determines that r9c8=6. Either way, r9c8=6. |
Is it beautiful? Is it terrible? Simple, yet elegant? Involved and ugly?
A matter of taste, isn't it?
---
Speaking of taste, did any of you people ever try Durians?
Last edited by nataraj on Tue Aug 05, 2008 7:19 pm; edited 1 time in total |
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wapati
Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.
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| Posted: Tue Aug 05, 2008 7:16 pm Post subject: |
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| keith wrote: |
Add a clue to puzzle A, making a harder puzzle, B.
Remove a clue from puzzle B, making an easier puzzle A.
Are these not the same thing?
Keith |
No. If you have a valid puzzle you can make any non-given cell a given, provided you put in the "correct" value and you will still have a valid puzzle.
Take a valid puzzle and remove any given and you most likely do NOT have a valid puzzle. To get your "Remove a clue from puzzle B, making an easier puzzle A" you must add a step that is essentially "check to see that you have a valid puzzle", and probably do that many times to find one that works.
The thread above was getting concerned over the removal when the original premise was the addition. |
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Marty R.
Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA
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| Posted: Tue Aug 05, 2008 9:09 pm Post subject: |
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| Quote: | | I think the 3s in col 7 are as far from a forcing chain as one can get. No chain at all, simply a (strong) link. No implication yet, only an observation: there are exactly two 3s - one of the two cells must be 3. |
We may have different definitions of forcing chains. To me, taking a house with two possibilities and testing for each is no different from looking at a bivalue cell and testing each value. Basically, looking at any "either/or" situation and checking both without knowing where it might lead is a trial-and-error forcing chain.
Spotting a pattern, such as a Finned XY-Wing, that directs one to a particular "either/or" situation might be one step up the respectability ladder, but it still ends with testing for each of two situations to see where it leads. |
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