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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Tue Mar 04, 2008 12:29 am Post subject: Medusa multicoloring 


Consider the following fragment: There is a skyscraper on 4 in rows 17 and a strong link in column 6. So one half of the skyscraper was colored AB and the other one CD. The 4 in r2c6 is colored A because of the strong link. Is the 4 in r2c8 a valid trap based on seeing both an A and D? If I correctly interpreted what I read in another thread, all color combinations are valid traps except AC, the colors in the weakly linked column.
Code:  ++++
 . . .  14C5 . .  4D5 . .
 . . .  145 . 24A9  . 145 .
 . . .  . . .  . . . 
++++
 . . .  . . .  . . . 
 . . .  . . .  . . . 
 . . .  . . .  . . . 
++++
 . . .  4A6B . 4B6A  . . . 
 . . .  . . .  . . . 
 . . .  . . .  . . . 
++++ 


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Asellus
Joined: 05 Jun 2007 Posts: 865 Location: Sonoma County, CA, USA

Posted: Tue Mar 04, 2008 11:38 am Post subject: 


Quote:  Is the 4 in r2c8 a valid trap based on seeing both an A and D? 
No. Cluster #1 is "AB", so A and B are conjugates. Cluster #2 is "CD", also conjugates. A and C are weakly linked ("bridging" the two clusters). This induces a strong inference between B and D (one or both must be true; they can't both be false). In link notation: B=AC=D > B=D.
So, in addition to the Cluster #1 "AB" traps and the Cluster #2 "CD" traps, you now have a possible "BD" trap between the two clusters. "AD" and "CD" are of no use that I know of (lacking any additional information).
When two clusters are bridged by a weak link, the induced strong inference is always in the two colors that are NOT involved in the weak link.
Note that it is also technically possible to bridge two clusters using a strong inferential link (using, say, wing pincers), though this isn't seen so often. In that case, the colors involved in the bridge perform the traps between the two clusters and a weak inference is induced between the two colors not involved in the strong link bridge. It is the mirror image, color wise, of the weak link bridge. (Link notation: BA=CD. The A=C bridge is necessarily strong, so the conjugates take on the weak link role in the alternating inference.)
If you stick to two clusters, that's all there is to it. However...
If you progress to three or more clusters, with various bridges, you must keep track of all of the induced strong inference pairs. As clusters expand, they often "see" other clusters in new ways, creating new weak link bridges and inducing new strong color pairs. (This quickly leaves the paper and pencil realm.)
New strong inferences can also be deduced from existing strong pairs if they share a conjugate pair. For instance, let's add a third cluster above, "EF", with a weak link bridge to the "CD" cluster involving D and F. The resulting induced strong pair is "CE". So we now have "BD" and "CE" as induced strong pairs. However, since C and D are conjugates, "BE" is also a strong pair and can perform traps between Cluster #1 and Cluster #3. (Link notation: B=DC=E > B=E, the conjugate "CD" pair again taking on a weak role.)
A suggestion: Use upper/lower case letters for the cluster conjugates (A and a for one cluster, B and b for another, etc.). It makes it easier to spot conjugates among the strong pairs and deduce new strong pairs. For instance, it is then easy to see that if you have "Be" and "CE" strong pairs, you also have a "BC" strong pair.
I answered more than you were asking. But, it gives you some more to explore. 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Tue Mar 04, 2008 4:48 pm Post subject: 


As always, thanks. You must realize by now that I have trouble with inferences, bridges, and the like. I have reached my skill levelwhatever that level might beby mechanical means. I don't see patterns and deduce very well. What I do is learn techniques and get them stated into "rule" form, then apply those rules to specific situations.
I obviously misunderstood/misinterpreted this statement in this thread:
"I simply note the colors in the margin: Eliminations can be made by red  green, blue  orange, AND by red  orange."
http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=2389
Again, thanks. 

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keith
Joined: 19 Sep 2005 Posts: 3165 Location: near Detroit, Michigan, USA

Posted: Tue Mar 04, 2008 10:14 pm Post subject: 


Marty R. wrote:  As always, thanks. You must realize by now that I have trouble with inferences, bridges, and the like. I have reached my skill levelwhatever that level might beby mechanical means. I don't see patterns and deduce very well. What I do is learn techniques and get them stated into "rule" form, then apply those rules to specific situations.
I obviously misunderstood/misinterpreted this statement in this thread:
"I simply note the colors in the margin: Eliminations can be made by red  green, blue  orange, AND by red  orange."
http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=2389
Again, thanks. 
Asellus wrote:  No. Cluster #1 is "AB", so A and B are conjugates. Cluster #2 is "CD", also conjugates. A and C are weakly linked ("bridging" the two clusters). This induces a strong inference between B and D (one or both must be true; they can't both be false). In link notation: B=AC=D > B=D.
So, in addition to the Cluster #1 "AB" traps and the Cluster #2 "CD" traps, you now have a possible "BD" trap between the two clusters. "AD" and "CD" are of no use that I know of (lacking any additional information). 
Marty,
To combine my example (colors) and Asellus' explanation (B=AC=D):
A = red
B = green
C = blue
D = orange
AB and CD make a skyscraper.
Keith 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Wed Mar 05, 2008 1:28 am Post subject: 


Quote:  Marty,
To combine my example (colors) and Asellus' explanation (B=AC=D):
A = red
B = green
C = blue
D = orange
AB and CD make a skyscraper. 
Keith, I'm just too confused. Your original statement said redorange (AD) was a valid trap, but my example was AD and Asellus said it was not valid.
I started out with AB and CD in the skyscrapers. I'm assuming that within each group the rules of both regular and extended Medusa apply. If that is true, what I need to know is what can generate traps and wraps. 

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Asellus
Joined: 05 Jun 2007 Posts: 865 Location: Sonoma County, CA, USA

Posted: Wed Mar 05, 2008 8:49 am Post subject: 


Marty,
There is nothing special about the particular colors. In Keith's example in the other thread, the weak link "bridge" between the two clusters happened to be green and blue. Green was conjugate with red, and blue was conjugate with orange. Therefore, the induced strong inference was between red and orange, the colors that were not involved in the weak link bridge of that example. However, in your example above, the weak link bridge is between A and C, or "red" and "blue", using Keith's assignments. So, in this case, the induced strong inference is between green and orange (B and D).
When he responded, Keith chose to assign colors to the ABDC letters you used above. Unfortunately, he didn't assign them in a way that matched his previous example. That is probably a good thing because it helps make it clear that there is nothing special about the colors one chooses. What is special is the links in the "bridges".
I hope that this bit of confusion ends up helping the concept make sense, because it really isn't that difficult. I am sure that you can see it, and that the "rules" will make sense to you. 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Wed Mar 05, 2008 5:46 pm Post subject: 


Asellus wrote:  Marty,
There is nothing special about the particular colors. In Keith's example in the other thread, the weak link "bridge" between the two clusters happened to be green and blue. Green was conjugate with red, and blue was conjugate with orange. Therefore, the induced strong inference was between red and orange, the colors that were not involved in the weak link bridge of that example. However, in your example above, the weak link bridge is between A and C, or "red" and "blue", using Keith's assignments. So, in this case, the induced strong inference is between green and orange (B and D).
When he responded, Keith chose to assign colors to the ABDC letters you used above. Unfortunately, he didn't assign them in a way that matched his previous example. That is probably a good thing because it helps make it clear that there is nothing special about the colors one chooses. What is special is the links in the "bridges".
I hope that this bit of confusion ends up helping the concept make sense, because it really isn't that difficult. I am sure that you can see it, and that the "rules" will make sense to you. 
Asellus,
You'd be surprised at what doesn't always make sense to me. I realize that there's nothing special about colors; in my regular and extended Medusa I use A and B because they just come naturally to me as opposites. Is it possible to reduce this multicoloring to simple "rules"? Or is it not that easy?
Am I correct in assuming that when starting two separate chains, that the regular rules apply within each group? The wraps of AA, Aa, a,a, BB, Bb and bb? And the regular traps of AB, Ab, aB and ab?
If the above is correct, then what? At the end of your first paragraph are you saying thatusing my exampleBD is the only valid trap between the two groups? If yes, would any combination of those in upper and lower case also be valid? Are there any valid wraps between B and D? 

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Asellus
Joined: 05 Jun 2007 Posts: 865 Location: Sonoma County, CA, USA

Posted: Wed Mar 05, 2008 9:57 pm Post subject: 


Marty,
The rules for Medusa multicoloring are really fairly simple. But first, we need to clear up some possible confusion.
When performing Medusa multicoloring, you must forget all about "extended" Medusa. The two things are very different. If you try to "extend" the color clusters when doing multicoloring, you will create too much complexity. You say you use "A" and "B" for your cluster, then you mention "a" and "b", so I assume those refer to the extensions of "A" and "B". So, let's be clear that Keith and I are not referring to such "extensions" in anything we've written above. The clusters we are talking about are strictly the conjugate clusters of basic Medusa.
When I recommended using Aa, Bb, Cc, etc., for marking the clusters, I did not mean to suggest using "extended" markings.
Let's turn to something louder than words:
Code: 
++++
 34B79 3C4c 459  1 6 8  2379 257 23579 
 368 2 568  7 4 9  368A 1 356 
 679 678A 1  3 5 2  679 678a 4 
++++
 3C8c 5 7  6 1 4  238 9 238 
 2 3c4C 4c6C  9 8 5  3C67 67 1 
 689 1 689  2 3 7  5 4 68 
++++
 5 67 3  8 9 1  4 26 267 
 14b78 9 4B8b  5 2 6  178 3 78 
 16 6A8a 2  4 7 3  169 568A 569 
++++

(This is taken from a solution I posted here. It is my step "[7]" in that post.)
The first cluster is Aa, the second is Bb, and the third is Cc. Each cluster is a basic Medusa cluster and is colored as far as the available conjugate links allow.
Now, look at b7. There is a weak link between 8a in r9c2 and 8b in r8c3. This serves as the "bridge" between the Aa cluster and the Bb cluster. Since the bridge colors are ab, the strong inference colors are AB. So, AB can perform traps between these two clusters. (There aren't any, however.)
Next, look at c3. There is a weak link between 4B in r8c3 and 4c in r5c3, bridging clusters Bb and Cc. The bridge colors are Bc so the strong inference colors are bC. bC can perform traps between these two clusters. (Once again, alas, there aren't any.)
Now, we examine those two induced strong pairs: AB and bC. Notice that they include the conjugate colors Bb between them. That means that AC is also a strong pair and can perform traps between the Aa cluster and the Cc cluster. This time, there IS such a trap... in r2c7. The <3> shares the cell with 8A and can see the 3C in r5c7. So, it can be eliminated.
There you have it. This example contains pretty much all there is to know about Medusa multicoloring. As an extra, you can see if you can follow the AIC in that other post using the colorings in the grid above. It should be easy to see how it works.
As another extra, note that there is also a Bc bridge in b1. That means that a more roundabout AIC could have taken that route. With Medusa multicoloring, there is no need to find the stepbystep AIC route. The coloring rules allow you to perform the eliminations without finding the AIC details.
I hope this has helped to make things clearer. 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Thu Mar 06, 2008 1:04 am Post subject: 


Quote:  You say you use "A" and "B" for your cluster, then you mention "a" and "b", so I assume those refer to the extensions of "A" and "B". 
Yes, those referred to extensions, which I incorrectly assumed could be done within each cluster.
I'd be lying if I said I understood this post. However, I've printed it out and will study it; I have problems following things on a monitor. 

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