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Captain Pete
Joined: 09 Jun 2007
Posts: 55
Location: Oley, PA
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 Posted: Fri Feb 15, 2008 3:15 pm Post subject: Menneske SH |
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Looks easy, but I'm stuck:
| Code: |
+--------------+-----------+-------------+
| 2459 456 1 | 8 24 59 | 256 3 7 |
| 2459 3 8 | 246 7 59 | 256 1 245 |
| 7 456 245 | 246 3 1 | 9 245 8 |
+--------------+-----------+-------------+
| 45 2 45 | 7 8 3 | 1 9 6 |
| 8 7 3 | 9 1 6 | 4 25 25 |
| 1 9 6 | 45 45 2 | 7 8 3 |
+--------------+-----------+-------------+
| 6 8 245 | 3 9 7 | 25 245 1 |
| 245 45 9 | 1 6 8 | 3 7 245 |
| 3 1 7 | 25 25 4 | 8 6 9 |
+--------------+-----------+-------------+
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Play this puzzle online at the Daily Sudoku site |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Fri Feb 15, 2008 3:46 pm Post subject: |
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coloring
2:-r3c3=r7c3-r8c1=r8c9-r5c9=r5c8-; r3c8<>2
and 4:-r2c9=r3c8-r7c8=r7c3-r4c3=r4c1-;r2c1<>4
both are not the usual xwing or simple skyscraper/kite. |
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Steve R
Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England
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| Posted: Fri Feb 15, 2008 4:30 pm Post subject: |
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After eliminating 2 from r3c8, following nataraj or using the X-wing (skyscraper) for 2 in columns 3 and 7 you have two W-wings on (45).
One is in r3c8 and r8c2; the other in r3c8 and r4c3. As luck would have it the conjugates for 4 in row 7 link both.
So 5 may be eliminated from r3c23 and the rest is plain sailing.
Steve
[Edited to change "column" to "row"]
Last edited by Steve R on Fri Feb 15, 2008 9:00 pm; edited 1 time in total |
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dejsmith
Joined: 23 Oct 2005
Posts: 42
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| Posted: Fri Feb 15, 2008 8:08 pm Post subject: |
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nataraj
Could you explain how you extended your coloring on 2 from r8c9 to r5c9? My limited knowledge of coloring prevented me from making that extension since r2c9 also has a 2? It appears that it is somehow related to the 25 pair in r5c89 since you colored on the 4 apparently using the same logic with the 45 pair in r4c13.
Thanks...Dave |
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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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| Posted: Fri Feb 15, 2008 8:40 pm Post subject: |
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| Steve, I get the 45 W-wing in R8C2 & R3C8 (because of the conjugate link in 4s in C9, not C7), but I don't see the 45s in R3C8 & R4C3 as related? Still works out OK I think. Nice puzzle, thanks! |
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Steve R
Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England
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| Posted: Fri Feb 15, 2008 9:03 pm Post subject: |
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Sorry, Victor.
I wrote "column 7" in error for "row 7." I have edited te original post in case anyone lse reads it!
Steve |
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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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| Posted: Fri Feb 15, 2008 9:08 pm Post subject: |
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| Sorry Steve in return - should have spotted that myself! |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Fri Feb 15, 2008 10:00 pm Post subject: |
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| nataraj wrote: | | both are not the usual xwing or simple skyscraper/kite. |
They are both instances of 2-cluster multi-coloring. Or, they are also called Color Wings.
For the <2>s, the two clusters are "bridged" by the weak link in c9 (which is why there can be other <2>s in c9). Similarly, the two <4> clusters are bridged by the weak link in c3.
Note that the eliminating ("trapping") polarities must be the opposite of those involved in the "bridge." nataraj assures this by using the alternating weak-strong link chain notation.
Here is one way to show the coloring:
| Code: |
+---------------+-----------+--------------+
| 2459 456 1 | 8 24 59 | 256 3 7 |
| 2459 3 8 | 246 7 59 | 256 1 245 |
| 7 456 2g45 | 246 3 1 | 9 -245 8 |
+---------------+-----------+--------------+
| 45 2 45 | 7 8 3 | 1 9 6 |
| 8 7 3 | 9 1 6 | 4 2R5 2G5 |
| 1 9 6 | 45 45 2 | 7 8 3 |
+---------------+-----------+--------------+
| 6 8 2r45 | 3 9 7 | 25 245 1 |
| 2g45 45 9 | 1 6 8 | 3 7 2r45 |
| 3 1 7 | 25 25 4 | 8 6 9 |
+---------------+-----------+--------------+
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The "RG" and "rg" clusters are connected by the bridge colored "rG". So, eliminations occur with "Rg". |
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ravel
Joined: 21 Apr 2006
Posts: 536
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| Posted: Sun Feb 17, 2008 12:11 am Post subject: |
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Hm, very deep explanation, Asellus, but i fear only insiders understand "bridged clusters".
So let me try to explain nataraj's elimination in chain words.
Either r3c3 is 2 or
r7c3=2 => r8c1<>2 => r8c9=2 => r5c9<>2 => r5c8=2
i.e. one of r3c3 and r5c8 must be 2 and r3c8 cant be 2.
When i go through the numbers, i look at strong links, and do something like "strong - weak - strong - weak - strong ..." in my head. Always after "strong" i look if the cells, which see the starting and this strong cell, contain the number.
Here starting with the strong link in column 3 to r7c3 its immediately clear, that the weak links to r7c79 cannot help you (no strong link from there). From r8c1 no strong link in column 1, but to r8c9 (but no 2 in r3c9 or r8c3).
Then again a weak link to r2c9 leads to nothing, but r5c9 connects a strong link to r5c8, which gives the elimination.
Note, that when looking for such strong link eliminations you can save a lot of time, if you know patterns, where no elimination is possible. They are similar to 2-value uniqueness patterns like (empty) x-wings or pairs "around the corner":
| Code: | x . x | . . x | . . .
. . . | . . . | . . .
x . x | x . . | . . .
-----------------------
. . . | x . x | . . .
. . . | . . . | . . .
x . x | . . . | . . .
-----------------------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . . |
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nataraj
Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
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| Posted: Sun Feb 17, 2008 5:47 pm Post subject: |
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| dejsmith wrote: | nataraj
Could you explain how you extended your coloring on 2 from r8c9 to r5c9? |
Dave,
I shall try to explain by using a picture (and hope it is worth a thousand words indeed). Here I plotted the "strong links" in 2, meaning (for the purpose of this explanation) all rows, columns and boxes with exactly 2 occurrences of the candidate "2" (and mark with "o" all other cells containing "2"):
| Code: |
+·····+·····+·····+
·o · * ·o ·
· · | · ·
·o ·o | ·o o·
· · | · ·
· *·o | · o ·
+····|+··|··+·····+
· |· | · ·
· |· | · ·
· |· | · *-*·
· |· | · ·
· |· | · ·
+····|+··|··+·····+
· *· | ·o o ·
· /-/ · | · ·
·*---------------*·
· · | · ·
· ·*-* · ·
+·····+·····+·····+
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In order to eliminate a candidate frm a cell, we are looking for cells that can see BOTH ends of a chain that is made up of an ODD number of links AND at least every other link must be a strong one.
The chain starts at r3c3. The first link must be strong (always). This is the link r3c3=r7c3. A strong link is indicated by the equal sign "=" in the notation I used in my original post.
The second link could be strong or weak, it does not matter (remember, only every other link needs to be strong). As it happens, the second link is also strong, but that is not essential - it is important, however, that there IS a link. This "weak" link is indicated by the dash "-" in the chain notation. It is the link between r7c3 and r8c1.
The next link needs to be strong again, it is r8c1=r8c9.
The next link does not need to be strong (this was your question - why am I using the connection between r8c9 and r8c5 when there is a 2 in r2c9. Answer: only every other link needs to be strong), it connects r8c9 and r5c9.
One more strong link needed, remember we need an ODD number of links, so we have r5c9=r5c8 and the chain is complete. It connects r3c3 and r5c8 and the cell that sees both ends (and has a 2 in it) is r3c8.
As ravel pointed out, the reasoning behind this whole chain business is quite simple: read every strong link as "if .. is not 2, then .. is 2" and every weak link as "if .. is 2 then .. is not 2". Taken as a whole, the chain asserts that if r3c3 is not 2, then r5c8 is 2 and thus r3c8 cannot be 2. |
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dejsmith
Joined: 23 Oct 2005
Posts: 42
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| Posted: Mon Feb 18, 2008 1:24 am Post subject: |
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Thanks to all for your great explanations. I will print them out for my solutions techniques notebook. I can't wait to attempt applying this new (for me) strong link/weak link coloring concept, as I have only considered strong links before. And, thanks nataraj for "using a picture". My mind still starts blanking out from seeing "if...then" due to too many logic courses taken in college many, many years ago.
Dave |
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