Nonpatternbased strategies  MTB KwonTom Addict Puzzles: 363 Best Total: 34m 58s  Posted  2007.03.30 22:17:56 I finally made Club 59 today, and after seeing the welcome message, I decided to check out the forums for the first time.
The real surprise here was that the method I've been using is pretty substantively different from what it seems like most of the folks here are using. I only look for a very limited set of patterns, and then do most of the rest based on a set of general principles I've derived from the game. (At least, until I get to a stopping point, and then I generally "Fix Position" and trialanderror it from there.) Now, since I've only just broken an hour, and there are solvers out there knocking on 10 minutes, I'm thinking I really ought to study what they're doing. On the other hand, this method has gotten me this far, and is a logically strict method, so I thought I'd at least start by sharing some tidbits. Please let me know if this is interesting, and I can add more. On the other hand, if other people are already using this, or if it's already discussed elsewhere on the forum, I'd love to know.
The easiest (and less useful) one I call the parity principle, and it's very similar to some other things I've seen discussed. I think I saw Foilman mention it as "make sure there's an even number of ends entering and exiting an area." I restate this slightly. Consider paths through a puzzle hopping from number or blank to number or blank, rather than from dot to dot, where you can't jump diagonally over dots, but have to jump over edges. The number of lines you cross in making a complete path will always be even. It's logically the same as above, but I find it easier to spot trapped areas this way.
The more useful one, which has pretty much driven how I do these puzzles, I call the "activity principle," for lack of a better way to think about it. Aside from it, the only real hard and fast patterns I use are the adjacent 3s, diagonal 3s, and, say, a 3 and a 1 adjacent on an edge. There's a few others I probably recognize, but I don't think of the puzzles this way. I think about them using the following rules:
For now, let me define an angle as two slots at 90 degrees to each other:
In the above diagram, the ?'s form one angle, as do the X's another, and the lines another. Naturally, any dot can be looked at as a collection of four angles.
Borrowing from geometry, define the ? angle and the X angle as "opposite" angles of each other.
Similarly define the line angle and the ? angle as "diagonal" from each other. (across the same square, but not sharing an edge.)
Now for the critical stuff. Define an angle as "active" if contains one line and one cross. Define it as "inactive" if either contains two lines or two crosses. Once you start to look at the puzzle this way, all sorts of stuff starts coming out.
For starters, the following rules can be deduced logically pretty easily. (I won't provide proofs to start with, but can if there's question about them.)
1. Any square (a number or a blank) consists of four angles. There will always be an even number of active angles in any square. (This is really where the name active/inactive came from originally. From the square's perspective, an angle is active if the loop enters/leaves the square through that angle, and inactive if it either stays in the square through that angle, bounces off, or doesn't touch that angle.)
2. Opposite angles will always have the same activity.
3. Diagonal angles in squares with even numbers in them (0,2, although the 0 is a trivial case) will always have the same activity. Diagonal angles in squares with odd numbers in them (1,3) will always have opposite activity.
Those are the fundamental principles, which don't get you very far. But it's not hard to deduce some rules which essentially get you many of the basic patterns talked about in some of the other threads. First, there's rather easily derived numberspecific rules:
0's:
Obviously, all angles within a 0 square are inactive. By simple extension of Rule 2 above, all angles opposite of the 0 angles (here marked with ?'s) are also inactive.
1's:
Inactive angles inside a 1 square are always doubleXes.
3's:
Inactive angles inside a 3 square are always doublelines. Opposite angles to the 3 square can *never* be double lines. (Hence the ? in the image must be an X, not a line.)
2's:
2 squares are where a lot of the fun happens. They're the most polymorphic, but have some really interesting properties. In general, they have two forms  parallel lines, or perpendicular lines. Parallel lines occur if and only if all four corners of the square are active. Perpendicular lines occur if and only if one set of diagonal corners are active, and the other set inactive.
The presence of any inactive corner on a 2 square (which can be implied by an inactive corner opposite to one of its angles) means it's the perpendicular form. This also means the two angles in the diagonal set not including the inactive one will be active. (I know this is starting to get abstract...)
There's more that can be derived, but I'll just do some patterns first:
This is probably familiar to the old hands as a pattern, but there are 2 linse and 2 X's here:
Here's a favorite of mine: 4 lines, 4 X's:
With the activity principle, you could have hundreds of twos diagonally between the 3s, and still find 4 lines and 4 X's.
And just to get wacky...
The state of all of the ?s are quickly identifiable using the principle here. There's probably more you could deduce from this pattern, but I could fill in the ?s in as fast as it took me to click them.
Do other people look at things like this similarly? I'm really curious... 
Tilps KwonTom Obsessive Puzzles: 4320 Best Total: 20m 22s  Posted  2007.03.31 00:41:29 Sounds like what I call coloring. Which is where you 'color' edges which are known to be the same the same color, and edges which are definitely opposite an opposite color. Once you have a coloring, you can extend coloring using numbers and intersections. A color anticolor pair at an intersection implies a coloranticolor pair on the other side of the intersection. Same for adjacent edges on a two. 
procrastinator KwonTom Obsessive Puzzles: 1083 Best Total: 12m 56s  Posted  2007.03.31 04:50:16
Quote: Originally Posted by mtb Do other people look at things like this similarly? 
Definitely. Well, it's among the first things I learnt, and I can't imagine anyone completing a Friday puzzle without it  or even most Mondays, probably. But we haven't established names for the concept. Maybe we haven't talked about it too much because it just doesn't lead to anything too complex, or maybe you've found some deeper insights to kick off such a discussion. Sometimes Jankonyex talks about numbers in a way that sounds very similar to your counting of "activity", but that could just be my imagination since I haven't really tried to understand those posts. When I one day work out how to read Jankonyex's unicode diagrams I'll go back through and decipher his impressive oeuvre.
Anyway, propogation of information along diagonals is complementary to patternspotting. Typically you will fill in a pattern, then trace it's repercussions along a diagonal to kick off another pattern at the other end. 
