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[Asellus]

On really tough puzzles, it is sometimes possible to make progress by extending Medusa Coloring in a fairly simple way. I'll explain the idea first, then offer some suggestions on applying it. (Note: I don't claim any originality here. I haven't read a description of this approach elsewhere, but that's most likely due only to my lack of research. It is akin to a watered down GEM and has surely been done by others.)

Start with normal Medusa using, say, "R" and "G". This may not extend very far or even provide any eliminations, but that's okay. Next, one or both of these colors is extended by assuming that it is true and marking resulting placements with alternate markings. For instance, "r" can be used for placements that result if "R" is true and "g" for placements that result if "G" is true.

It is important to note that the "r" and "g" coloring is not Medusa. It is not valid to exploit bivalues, for instance, in r-g marking. This should become clear in the examples.

The rules for using these color marks are fairly simple:

(1) Red-green "trap" eliminations are possible between any red-green combination EXCEPT "r" and "g". So, R-G (normal Medusa), r-G, and R-g are all valid "traps"; but r-g is not a valid "trap". [Edit: This Rule is too restrictive. See the next post below for a correction.]

(2) Same-color "wrap" contradictions are valid in all cases. HOWEVER, only the original Medusa color is proved false; the related inference color is not determined by a color "wrap". In other words, G-G or G-g or g-g for a single candidate in a single house or cell is a "green wrap" and means that all "G" values are false and all "R" and "r" values true. But, it says nothing about the "g" values, which would need to be erased before proceeding if the puzzle were not solved. [Edit to restate this rule more accurately.]


As far as deciding how to apply this approach, I suggest looking for and trying to exploit potential locked sets for finding useful chains of inferred "r" and/or "g" values. It helps if the normal Medusa colored portion is a bit extensive. But, that isn't always necessary or possible.

The nice thing about this approach is that it's not really much more difficult than basic Medusa, which isn't very difficult. But, it's quite a bit more powerful.


Here is an example using a puzzle that was posted on a neighboring thread:

[Asellus]

After some discussion in that alternate thread, I realize that my "Rule (1)" is too restrictive: r-g pairings are allowable for trapping as well; no exception is necessary. This is good news since it makes this coloring approach even easier and more powerful. Here is the restated rule:


(1) Red-green "trap" eliminations are possible between any red-green combination. In other words, R-G (normal Medusa), r-G, R-g and r-g are all valid "traps."


Applying this in the example above allows progress to be made more easily. For instance:

[Asellus]

Exploring this approach further, I finally found instances of something I suspected might be possible. It deserves being mentioned as a third "Rule":

(3) If an instance of a candidate digit is inferred to be both "r" and "g", then it must be true and can be placed immediately.

I will work of preparing an example to post.

[Asellus]

More regarding "Rule 3":

In going back over possible examples, I discovered that any time I found a candidate digit that must be both "r" and "g" there was one (or more) "Trap" eliminations nearby that accomplished the same thing by eliminating the alternatives. So it seems that, technically, "Rule 3" isn't necessary. However, it makes a nice short cut "rule of thumb" in actual practice since one sometimes spots a "this must be both red and green" candidate before one spots the Traps that force it to be so. (Setting the "both red and green" candidate True eliminates the candidates that would otherwise be Trapped. So, it's two ways to see the same thing.)

[Edit to remove text that described something that is not logically possible and would indicate that one had made a coloring error.]

Meanwhile, I continue to play with this technique on very difficult puzzles and find that it has been able, so far, to solve them fairly easily. For instance, the 8-Jan-2008 Daily Nightmare at sudocue.net can be solved entirely in this way. Otherwise, the only "traditional" technique apparent is a very difficult to spot Sashimi X-Wing that accomplishes little. After that it seems you have no choice but to dig deep! Among possible "deep" tools, this extended approach to Medusa is able to solve it without too much difficulty.

[Marty R.]

Asellus,

I want to see if I can tiptoe into this thing without having much of an understanding of AICs, ALSs, Eureka notation, etc. There have been a lot of posts with a lot of examples and I'm trying to see if I can distill this thing down to a manageable size. Here's what I think I know.

I think I'm reasonably proficient at regular Medusa, or what is often called 3D Medusa.

I think "trap" is another term for what I'd call an elimination from a pincer situation.

I think "wrap" is another term for what I'd call a contradiction, such as two candidates in the same house with the same color or two numbers in one cell with the same color.

I've got the three "rules" written down, although I'm not sure if more can be done with the inferences over and above what is specifically stated in the rules.

Once you assume R or G is true, you can extend either with small r's or g's indefinitely. You can never place another capital letter except under the regular Medusa rules. But I assume regular Medusa can be extended if made possible by eliminations/placements from the three rules. Then you could start in with new assumptions.

[Asellus]

Marty,

I believe you've got it. As for that 8r in r9c2, I can no longer see how I came up with it. However, it isn't used in any of the subsequent steps I illustrated, so let's just pretend I didn't mark it! My apologies for the confusion.

The nice thing about this technique is that it requires no knowledge of AICs or Eureka notation. As for ALSs... they can come in handy in a couple of ways when extending a color. For instance, in the grid we are considering above the green <1> in r4c1 converts the {137} ALS in r12c1 into a green {37} locked pair, as mentioned. This would have rendered the <4> in the {347} cell r8c1 green if there hadn't been the strong link in c1 with that 4R.

Additionally, any ordinary elimination technique can be exploited in extending the coloring. For instance, assuming red to be true might create three "red" bivalue cells that form an XY-Wing that would eliminate a candidate somewhere and leave only one possibility for red in a victim cell (which could be marked "r"). These sorts of complex ways of extending a color are only needed in the most difficult puzzles, but are worth mentioning.

So, one might even be able to find, say, a color-inferred "xz" ALS that allows the extended coloring to extend further. (I'd be impressed by someone who posted an example of such a thing!) Or, maybe a "red UR" might do the trick. Or maybe something else. This is where this approach can get very powerful in those really super-tough puzzles.

[Edit to correct "X-Wing" to "XY-Wing".]

[Marty R.]

A,

Thanks very much, your assistance is always appreciated. I'm hoping that maybe this technique will help me solve some of the puzzles that now force me into T&E. I'm wondering how to do the markings for Ext. Medusa. For the regular, a blank grid works fine, not sure if it'll work for this.

Thanks again.

[Asellus]

In terms of P&P marking, that depends on how one does ones marking. The single digit grids, while very useful, don't help for Medusa. I usually use a dot grid within boxes for marks. This allows for some flexibility, but might seem obscure to many. For instance, a cell that has candidates {123679} would look like this:

[Marty R.]

It didn't take me long to figure out that the blank grid that works so well for chains and Medusa won't work, at least for me, for the extended version. I don't know yet if I can use my regular grid for markings and salvage it if things don't work out. It seems the ideal solution is an impractical one: take my grid to the library and make a copy. Oh well, I'm sure something will work as I think about it more. I don't think the dot thing would work for me. Maybe it will require making a second grid by hand, saving the trip to the library.

[Marty R.]

Update:

I either screwed up and got lucky or else this thing worked on the puzzle I was stuck on. And I was able to do it on the blank grid, although things weren't real complicated with large number of polyvalued cells.

I do have a couple of questions which I hope are quick and easy:

1) Can anything be done based on one number in one cell having both an upper and lower case letter? For example, suppose the cell is 5Gg6R. This phenomenon occurred three times very quickly.

2) Can an unlimited number of assumptions be made? If there are, say, eight Rs on the grid, can I start an assumption from each R such that the grid would be heavily populated with small r's?

[keith]

[Asellus]

[Marty R.]

A,

Once again, thanks. I hope the pace of my questions will quickly decelerate.

[Marty R.]

[Asellus]

[Marty R.]

Once again, thanks. I hope my questions are done, as I think I've covered the situations that I can anticipate. Having both forms of Medusa available to me is sort of empowering.

Would you be kind enough to also add a short reply when you edit the first post, since there is no other indication when an edit is made.

[Asellus]

I think I've got it all correct now! Let's hope there's no more edits.

[Marty R.]

This is about "promoting" a candidate from lower to upper case. I had a cell that contained 2g7 as a result of the 2 being inferred from a previous move. I was thinking about promoting the 2 because the 2 was a strong link in its row. But I checked and that would've led to an invalid solution.

Can we say as a generalization that a promotion can occur only after a strong link opens up as a result of eliminating excess candidates in a house?

[Asellus]

[Marty R.]

Once again, thank you sir. You'll note that I went 10 days without a question. I think I'm catching on!!