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New (?) Highlander-esque strategy
djpohly
Kwon-Tom Addict
Puzzles: 387
Best Total: 25m 1s
Posted - 2007.09.13 18:50:26
I've not seen this posted here; my apologies if it has.  Just figured I'd share it, since it often helps me close out a nearly-finished Loopy game--or occasionally a weekend KL--with a minimal amount of time spent thinking.  When playing Loopy on the insano-big setting, it's common to run into a pattern similar to the following (this is a very minimalized example):

Four lines, four crosses (aside from the six "trivial" crosses that could be marked around the 3s):
As is the usual convention, question squares can be anything, and blank squares must be blank.


Explanation/Spoiler:

The 3s on the sides can be any two peninsular (i.e. nearly-closed) loops, or you can think of them as one doubly-peninsular loop (nearly closed, but with two holes).  The Highlanderish 2s on the top and bottom of this diagram can actually be any isolated pattern which could be solved in two ways: one which connects two ends from the loops, and one which brings both ends through without connecting them.

By a fairly straightforward application of the Highlander argument, it can be deduced that neither of the two pairs of loop-ends connects.  Reasoning: if one connects, the other cannot, because that would create a closed loop.  But if one pair connects and one does not, an equivalent solution exists where the opposite pair connects instead.  Therefore neither pair can connect.

Towards the end of a puzzle, I'll often have several loops which are nearly closed except for "four-way stops" around a 2 or a blank square, and I can apply this to any that have two such holes.
procrastinator
Kwon-Tom Obsessive
Puzzles: 1083
Best Total: 12m 56s
Posted - 2007.09.14 00:45:50
Quote:
Originally Posted by djpohly
I've not seen this posted here; my apologies if it has.

I'm reasonably certain I've posted a clunkier version, but I've no idea where - I couldn't find it in "Logical Thinking". But anyway, you've created a much simpler archetype than me, so it's a refinement rather than a duplication.

The danger with this pattern is that the only way for information to propagate from this pattern to the rest of the puzzle is if one of the halves you've made comes close to joining up to itself on the outside. Otherwise you're doing some pretty complex checking to verify the pattern then having to solve the outside independently, whereas if you'd just gone ahead and solved the outside first, this part would have come for free. So just because it's there doesn't make it profitable, though when it is profitable it can be terrific. I've only used it profitably once in a live puzzle, but I'm not particularly looking for it...

A nice (more common, I think) variation is when the 3 part (which is not necessarily just a 3, remember) is closed but not yet defined. Then the forced double-path through the centre can help define the 3 part, which can in turn feed information to the outside. I see this much more often.
Last edited by procrastinator - 2007.09.22 02:41:33
djpohly
Kwon-Tom Addict
Puzzles: 387
Best Total: 25m 1s
Posted - 2007.09.14 01:02:02
Yeah, this kind of thing seems to come up a lot more on Loopy than here.  Maybe a difference in the generation algorithms accounts for that, or maybe it's just the fact that Loopy puzzles can be so huge that it's not as easy to see all of the near-loops at once.

You have a good separate point there too... I've never really given much thought to determining what parts of a puzzle will "solve themselves" when a more important piece has been solved.  Seems a worthwhile chapter to add to my mental catalogue.
Naivoj
Kwon-Tom Addict
Puzzles: 314
Best Total: 33m 50s
Posted - 2007.09.15 02:49:23
Quote:
Originally Posted by procrastinator
A nice (more common, I think) variation is when the 3 part (which is not necessarily just a 3, remember) is closed but not yet defined. Then the forced double-path through the centre can help define the 3 part, which can in turn feed information to the outside. I see this much more often.
Procrastinator could you please post a basic example of your variation, as I don't understand your explanation. With a sample to work with, I usually can figure out most patterns/variations/stategies.
Thank you
procrastinator
Kwon-Tom Obsessive
Puzzles: 1083
Best Total: 12m 56s
Posted - 2007.09.22 03:07:09
Quote:
Originally Posted by naivoj
Procrastinator could you please post a basic example of your variation, as I don't understand your explanation.

I can't make sense of what I wrote anymore. Maybe it never made sense? I suspect that I was confusing aspects of DJPohly's pattern with aspects of the following one, and not realising the hybrid didn't actually work. The following pattern does work, and does qualify as easier (more local) to spot than _useful_ instances of DJPohly's one, but if this is what I was trying to describe then I failed miserably. It's not really a variation to begin with, though I'm sure you can appreciate the similarities.

Naivoj
Kwon-Tom Addict
Puzzles: 314
Best Total: 33m 50s
Posted - 2007.09.23 19:34:33
Quote:
Originally Posted by procrastinator
The following pattern does work, and does qualify as easier (more local) to spot than _useful_ instances of DJPohly's one, but if this is what I was trying to describe then I failed miserably. It's not really a variation to begin with, though I'm sure you can appreciate the similarities.
I understand this example and I can see it is following the same concept. There is one unique way for the inside path to get out of the 3 (a line below the 3 is forced making the 3 inside path). I think that the only squares that must be blanks in this pattern are left and right of the 2. Right and left of the 3 can be either blank or 2.

These highlander-esque deductions are more frequent in harder puzzles, like in the user puzzles.
For example this one which you can found in #208 "Bull's Eye Target":

12 Lines and 6 CrossesThere is one unique way to solve this corner area

Note that the following regular highlander 2 derived pattern can be applied first to my previous example:
2 Lines

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