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#1: Gator (Gator) on May 16, 2009 [HINT]

Edge logic required. Mwhahahaha...

Didn't have to guess at all.

As if anything else could be done, Gator.

I enjoyed this despite my not liking puzzles where I have to "guess" lol! The difference is...this is a 15x15...which makes the tiresome "gussing" managable and fun!

Well, I designed it to give people practice on edge logic. Small enough not to get overwhelmed too much.

Found to be logically solvable by jan.

I did it by asking where the 4 in R14 could be given that there was nothing but 2's in R14. This let me dot a lot of squares in R14, but didn't let me mark any blacks. Then I did a little thinking about where the "2" in C1 could be, given the "1" in C2. This let me dot one more cell in R14, and suddenly all sorts of stuff began to fill in.

A good exercise in slightly fancier edge logic, but I've seen better pictures.

I re-solved this one, starting with trying to place the "4" in R 14, and I was actually able to place 3 out of the 4 black pixels, and then it solved easily.

Here's what it looked like after the first step I took, doing edge logic on R14:

The "4" can't be in any of the dotted cells because if any part of it was in any one of those, then we'd have a "3" block in R13. It's not hard to see that the "4" can't actually be in the first few columns, but that isn't what I call edge logic. That requires quite a bit of look ahead. My step after this was observing that the "2" in C1 couldn't be in row 15 or 14, because that would place a black "2" near the bottom of C2, where it can't be. That eliminates the leftmost position for the "4" and the rest is not hard.

That's basically what I said, too, which is how I got 3 out of the 4 blacks in R14. Because of the last numbers in C's 1,2,3,4 being (in order) 2,1 3,3, the 4 can't go in the first set of blanks. I counted that as edge logic when placing that initial 4.

Comment Suppressed:Click below to view spoilers

I don't believe in "diagonal logic". I know there are certain recognizable patterns of clues that usually indicate some kind of diagonal line, but I haven't been able to see a way to prove that they can't also be created by a scattering of blobs.

I first got the bottom 2 and then ... piece of cake.

I started with column 1 figuring where the 2 could or could not go in relation to column 2. From there the rest just solved its self.

That was fun. I wasn't sure where to start with the edge logic, but when I figured out Row 14 (I did it like Adam described), the rest solved easily.

@Jan
Granted, there is a certain amount of analysis required before determining if "diagonal logic" makes it logically solvable. I've yet to find a situation where the puzzle has a single solution and "diagonal logic" doesn't (at least help) come up with the correct solution. You could just put it in the same category as any logic resulting from symmetric clues (they only work if there is exactly 1 solution, and you can ignore them in most cases without losing anything, but they assist in those cases where massive guess-and-check comes to the same solution).

In a previous life, I used to write long papers full of mathematical theorems and proofs, and a part of me still doesn't trust anything I can't prove. The proof that symmetric clues mean a symmetric image if the solution is unique is easy. The logic is all there. The only problem with it is that "if the solution is unique" clause. We can only use that logic to find a solution if we already know something important about the solution. That's the fatal flaw of symmetry logic. It's not that the logic isn't sound, it's that it depends on something you can only know from information external to the puzzle itself.

The problem with diagonal logic is that there is no proof at all. Certain kinds of patterns of clues suggest a diagonal line of cells. But what exact patterns of clues is it? Some patterns do mean there has to be a specific diagonal line. Some patterns allow either of two different diagonal lines. Some patterns allow solutions that look nothing like a diagonal line. Which is which? It may be possible to figure out a rule for these things, but I don't know how. It'd be cool if something could be found.

You make some good points, and I haven't spent that much time on proving the logic to be able to determine for any (arbitrary) case if the logic applies or not. I did prove it (not very formally, though) for when the line is 1 cell wide (some of the clues are '1's) in the thread where I first explained it - you can't end up with blobs, you just need to determine in the particular case if it goes one direction or the other; but for thicker lines (or lines of varying thickness) it isn't so trivial because of the possibility of getting blobs instead of a diagonal. There must be some more information given in the clues which "anchors" the diagonal in place. Then even after you know the general orientation of the diagonal(s), the question still remains of what exactly that tells you about the cells in the puzzle. That question is one which still needs answering (and currently still requires 'logic' in the form of guess-and-check).

I'll stop mentioning about diagonal logic in these puzzles unless I can explain how exactly it Proves that some particular cell is a particular color (which I admit is something I haven't done in any of my posts mentioning it - I've really only mentioned that it helped in getting to the final solution).

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