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Odd corners, even corners, and propagation
LoopGuy
Kwon-Tom Obsessive
Puzzles: 761
Best Total: 45m 59s
Posted - 2013.08.04 07:40:58
I've started to think about all this, and so I'm writing it down so people can comment on it.  A lot of this comes from Jankonyex, but I've taken my own path now and then.

Take a square.  It may or may not have a number in it.  Each corner of the square is made up of two sides.  These sides ultimately have 0, 1, or two lines.  Corners with 1 line are "odd" corners.  Otherwise, they are "even" corners (and, yes, zero is an even number).

For sake of this discussion, we will presume some of the blank squares actually have numbers in them, so we don't have any useful highlander clues.

This square has all even corners (I've marked the squares in question with a ?):



These squares have all odd corners:



And these squares have two odd and two even corners:



Now, let's define the "opposite" square as the one that shares a corner but not a side.  The two ?'s in the following are opposite squares:



Opposite squares share the oddness or evenness of their shared corner.  Here is an example of an odd corner.  Exactly one of the ?'s must be a line, but not both.  We say that the "oddness" propagates from one square to the next:



Even corners also propagate to the opposite square.  The ?s must either be both a line or neither can be a line:



The corners for a zero are always even:



Ones and threes always switch the evenness/oddness of their opposite corner (and, in fact, if you know a corner is odd or even, you can draw some x's or lines in the opposite corner with a one or three):



They also must have two odd and two even corners, so once you have two adjacent corners defined for oddness or evenness, you have that square defined.

The twos are where the tricky part lies.  Twos transfer the oddness or evenness to their opposite corner.  Furthermore, two of the corners must be odd, so if you get one even corner, you know what all the corners are.  If you get one odd corner, you know that the opposite will be odd, but you don't know if the other two corners will be odd or even.

For sake of illustration, I will end rows of diagonal numbers with 1's.  An odd corner for a one results in the opposite corner having x's, while an even corner for a 1 results in that corner having two x's.  For example (odd, then even):



So here we have a zero transferring an even corner into a chain of 2's.  We can put crosses on all the 1's at the ends of the chains:



Even a blank square can transfer oddness or evenness if it is restricted on two corners:



The two ?s can be marked with x's here.

Since 1's and 3's flip opposite corners, and must have two odd and two even corners, the line of propagation can "bend" if some corners are known:



In this case, I could just put the lines in the x's in the middle one and not have to think so hard.  However, what if I was just considering things:



Here, if the ? was an x, we get this:



So the ? must be a 1.

That's as far as I got.  Is this right?  Besides highlanders (and memorizing patterns for speed), am I missing any other truths?  Comments?
arbor8
Kwon-Tom Obsessive
Puzzles: 3074
Best Total: 14m 2s
Posted - 2013.08.05 10:17:50
Considering propagation, I tend to think with simple rules of what I call "diagonal charging".Diagonally, numbers pass the charge( full or
empty) , with following examples:




(not unique)



and if diagonal chain is unbroken, we can have like:

MondSemmel
Kwon-Tom Obsessive
Puzzles: 3798
Best Total: 7m 47s
Posted - 2013.08.05 16:01:37
"That's as far as I got. Is this right?  Besides highlanders (and memorizing patterns for speed), am I missing any other truths?  Comments?"

Yes, I'd say that's right. It's a very important rule for solving puzzles above a certain difficulty level. I already knew about it, but thanks for putting it in these terms. I read Jankonyex' explanation about it, but I find yours easier to understand. That said, I think about the matter more like arbor8, I think.

I would say this rule, plus patterns, plus highlander, plus the "any area must be entered by an even number of lines" rule (do we have a better, more descriptive name for that?), are all you need to solve any slither link puzzle. Well, really complicated puzzles require lots of trial & error exploration, too. Though the need for exploration decreases as one picks up more experience.
[As a side note, except for specific patterns, all these rules hold even for non-rectangular slither link puzzles, provided they have unique solutions.]

Also, I just noticed that the Wikipedia article for Slither Link actually contains lots of explanations as well. Might be a good, easy-to-comprehend resource for beginners. Did someone active on kwontomloop.com write part of this?
Last edited by MondSemmel - 2013.08.05 16:01:57
arbor8
Kwon-Tom Obsessive
Puzzles: 3074
Best Total: 14m 2s
Posted - 2013.08.05 16:45:12
I haven't yet found a better name to describe the rule " even number of lines entering an area". Important tool, but does anyone have a short but descriptive name for it?
Zyntax
Kwon-Tom Obsessive
Puzzles: 4006
Best Total: 13m 6s
Posted - 2013.08.05 23:50:55
The discovery of this 2's propogation method as you call it was the sole trick that allowed me to finally move from Club 29 to Club 19. It literally just hit me one night - a revelation if you will.

The way I thought of it was, if you have a line going "into" a 2 on a corner, then you have to have a line coming "out" of a 2 on the opposite corner. You've shown this above with diagrams, but that's how I describe/think of it in words.

For the people that are stuck in Club 29 (and higher) - I recommend you learning this trick. It shaved off considerable time for me!
LoopGuy
Kwon-Tom Obsessive
Puzzles: 761
Best Total: 45m 59s
Posted - 2013.08.06 01:34:29
Quote:
Originally Posted by MondSemmel
I think about the matter more like arbor8, I think.

So do I, honestly, and I suspect that is the most common application.  I have about 5 user puzzles left of all 1's and 2's, so I'm going to try the other bits out in those and see if it helps.
LoopGuy
Kwon-Tom Obsessive
Puzzles: 761
Best Total: 45m 59s
Posted - 2013.08.06 01:41:22
Quote:
Originally Posted by Zyntax
The way I thought of it was, if you have a line going "into" a 2 on a corner, then you have to have a line coming "out" of a 2 on the opposite corner.

Yes, those are the words I use as well.  I tried the even/odd language to see if there was anything else I was missing (and there was).

Quote:
Originally Posted by Zyntax
For the people that are stuck in Club 29 (and higher) - I recommend you learning this trick. It shaved off considerable time for me!

Well, this isn't my reason I'm in club 49, but I think it is because I'm spending too much time deducting patterns instead of just memorizing them.  If my average Monday is still in the 2 to 3 minute range, I'm not getting in Club 19 for awhile.
Tilps
Kwon-Tom Obsessive
Puzzles: 4138
Best Total: 20m 22s
Posted - 2013.08.06 09:15:27
Quote:
Originally Posted by Zyntax

For the people that are stuck in Club 29 (and higher) - I recommend you learning this trick. It shaved off considerable time for me!

I have known this 'trick' for years, club 19 remains a distant dream.  I just can't control the mouse fast enough.
MondSemmel
Kwon-Tom Obsessive
Puzzles: 3798
Best Total: 7m 47s
Posted - 2013.08.06 14:22:03
Quote:
Originally Posted by Tilps
I have known this 'trick' for years, club 19 remains a distant dream.  I just can't control the mouse fast enough.

Mouse speed is definitely helpful. I played RTS games for years (e.g. Warcraft III, Starcraft), and my mouse speed is correspondingly very high.
That said, Darklady apparently uses something like a tablet to solve these puzzles (sadly, I can't find the thread where he/she wrote that). So if mouse speed is an issue, have you considered trying a tablet?
Zyntax
Kwon-Tom Obsessive
Puzzles: 4006
Best Total: 13m 6s
Posted - 2013.08.06 16:36:20
Yeah recognizing recurring patterns is key. You'll just have to keep playing puzzles over and over until you see them, and then have them sink into memory where it's second nature to start placing segments on the board.

I've been playing these puzzles since 2003, but I didn't discover this site until late 2006. It was here that I've ramped up my skills in solving these things, and I think that's because it's more competitive here, and the difficulty of these puzzles were quite a bit harder than I used to see.
Darklady
Kwon-Tom Obsessive
Puzzles: 3489
Best Total: 9m 37s
Posted - 2013.08.06 19:52:11
Quote:
Originally Posted by MondSemmel
I played RTS games for years (e.g. Warcraft III, Starcraft), and my mouse speed is correspondingly very high.

That explains a lot!

I use a drawing tablet, and while it's probably not any faster than MondSemmel's mousing speed, it's also probably a lot easier to get up to that speed with a tablet than with a mouse. Also, in my experience, it's a lot more comfortable. Tablets are simply far more suitable for drawing lines than mice are, though Slitherlink puzzles require simple enough lines that mice are okay too.

Now, I did buy it and use it primarily for my art hobby - it just happens to be handy for things like this as well. They can be a bit of an investment, but if you wanted one primarily for this, I'm sure even a cheap old one would be fine.

(Also, they do take a bit of time to get used to, if you've never used one before - the disconnect between drawing on a blank surface and having the lines appear on screen makes it a bit trickier than drawing on paper. I guess there are touchscreen tablet computers out there now that wouldn't have that problem, and would probably have fine enough control for these puzzles... but I don't know about those.)

(And, yes, "she". Isn't that obvious?)
MondSemmel
Kwon-Tom Obsessive
Puzzles: 3798
Best Total: 7m 47s
Posted - 2013.08.07 09:48:34
Thanks for elaborating on your use of a drawing tablet! As I said, I couldn't find your original post anymore, so I couldn't suggest any specifics to others who might benefit from using such a tablet.

(And I apologize about the he/she mishap. In my "defense": TV Tropes.)
Last edited by MondSemmel - 2013.08.07 09:48:44

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