Here are pictures for three of the mentioned more advanced techniques that are the least understandable from my descriptions. See if you can figure out what's forced in each, and which of the verbal descriptions they match.

- slalom-1.png (3.26 KiB) Viewed 272365 times

- slalom-2.png (1.66 KiB) Viewed 272365 times

- slalom-3.png (2.53 KiB) Viewed 272365 times

I over-complicated the examples a bit to throw in demonstrations of more common patterns as well, but every line that can be determined in them requires use of the primary rule being demonstrated. In fact, I would really have trouble seeing how to apply the pattern in #3 although the pattern itself is quite obvious. The example started out larger and much easier, but I couldn't help myself shrinking it to the extreme. This is a puzzle forum after all, so there's a puzzle for you!

- slalom-1.png (3.26 KiB) Viewed 272365 times

This deduction represents "many sets of known-to-be identical cells interacting".

Mark the four squares between the 1-clues TL, TR, BL, BR from {top,bottom}-{left,right}.

- The 3-clue gives TL=BL, i.e. the lines in both left-side squares have to be in the same direction, to give 1 connection to the 3.
- Similarly, the top 1-clue gives TL=TR.
- Similarly, with 2-clues extending deductions, the two 2-clues on the right give TR=BR from the fact that the two squares to the right from TR and BR also have to be mutually parallel to fulfill the rightmost 2-clue.
- Now we have BL=TL=TR=BR. From BL=BR, we know that the 1-clue gets its one connection from one of those squares, therefore the lines in both squares below it must be away from the clue, i.e. \/.

See how the puzzle fails if a line is drawn to the bottom 1-clue from either square below it.

- slalom-2.png (1.66 KiB) Viewed 272365 times

This deduction represents "forming a 'dead-end U' next to a 1". The pattern also appears in the German tutorials mentioned above.

Mark the squares above the given "\/" L and R.

If R were "\", L would also have to be "\" to avoid making a closed loop. However, the rest of the lines around the 1-clue would have to be away from it, forming a larger loop around the line in L. This can also be seen as "the path can't continue through a 1-clue". Since R being "\" violated the rules, R must be "/".

This is just one example of the impossibility of forming a "dead-end U" opening towards a 1-clue. It can be even harder to spot if the line missing from the U is the "bottom" one. From this diagram, if R was already marked "\" and the square below it was blank, it would be similarly deduced as "\".

- slalom-3.png (2.53 KiB) Viewed 272365 times

This deduction represents ones "concerning possible paths for a shape to connect outside (impossible "around" a 3 for example)".

It's not necessarily all that advanced, but one that's needed more rarely. Thrown in this puzzle are more common deductions in "escape paths".

Starting from the intersection to the right from the 3-clue (top-left from the 1), a path must eventually be found to connect to an edge of the puzzle. Otherwise there would be a closed loop around either the intersection or the set of lines connected to it.

- The main deduction is simply that the path can't escape towards the bottom-left because it would have to go "around" the 3-clue, leaving it with only two connections.
- A more common deduction also needed is that it can't escape through the 1-clue either. Doing that would require (at least) two lines to connect to the 1-clue.
- There's nothing stopping us from drawing one line towards the bottom-left and/or to the 1-clue, but since they can't go further, there must be a line to the top-right. I.e. the top row of full squares ("\/ " in the puzzle) has to be completed to "\//".

See what the puzzle looks like if the deduced line was drawn "\" instead.

Obviously each pattern has possible variations, more complicated or just different. And of course there are extra clues around the patterns in real puzzles, including some to have already forced the lines given in my pictures.

One whole family of deductions I didn't mention earlier involves using the fact that a puzzle has a unique solution. I don't really like to use them, and there must be loads of them that I haven't even thought about. Anyway, a simple one is "/ /" against the top or bottom edge, with the added constraint that none of the four corners of the blank cell have clues in them. Often the deduced line (along with how it was deduced) will start a chain of deductions based on uniqueness.

Heh, beginners alright! I love that I haven't thought about the very last pattern of the third page there, the one with three 1-clues. I would see it with a clue in the center intersection, but not naked like that. The associated puzzle is very nice as well, pretty difficult but with small deductions.

In fact, there's a quite general rule in there that I haven't fully used: You need to have a path from every intersection to the edge, not just from clues or existing line segments. An intersection with no lines connected to it would immediately have a loop formed around it.

Edit: Added solutions to the mini-puzzles.