Slalom (aka Gokigen)

[nickdeller]

Draft rules text, inspired by but largely altered from that at wpcstylepuzzles.com:

Draw a diagonal line in each square of the grid. Numbers at some intersections show how many diagonals meet at that point. The diagonal lines must never form a closed loop.

Slalom example.png (11.43 KiB) Viewed 4359 times

Puzzle is known as Slalom in Germany and Gokigen in Japan, but there seems to be a wish for a more suitable English-language name. All ideas welcome!

[Mephisto]

"Slalom" may be a bad name for "Gokigen Naname". "Naname" means something like "obliqueness". Simon Tatham translated "Gokigen Naname" to "Slant".

Moreover, there is another Nikoli puzzle type called "Suraromu" (スラローム) - and "Suraromu" really means "Slalom".

See http://www.janko.at/Raetsel/Nikoli/Suraromu.htm

[PuzzleScot]

I agree - like in skiing, 'Slalom' implies travelling through a series of unaligned gates. This puzzle doesn't do that, so 'Slants' would be an improvement, but even then there should be a better word - 'Diagamaze'?

Reminds me of 'Hall of Mirrors' seen at fairgrounds - how about that as a name? That name would imply the 'no closed loops' part of the rules.

[david mcneill]

Nick,

The example works fine. In the rules, I would suggest changing the word "can" to "must".

David McNeill.

[nickdeller]

Thanks all. I'll leave the heading as 'Slalom' pending the perfect name emerging, and I think it needs to remain in the 'also known as'. As we have prior published examples using 'Slants' as the English-language name though, I reckon that's probably the front-runner.

[sknight]

I see once again by the high score table at Croco-Puzzle that a lot of folks still have real trouble with Slalom.
I thought I'd put up a few general tips. Feel free to add more.

Obvious break-ins include 0's and 2's along an edge, 1's or 1's in a corner, and 4's in the middle of a puzzle (you can't have a 0 in the middle...) as they are completely determined.

A general strategy that often provides a break-in is very similar to working along diagonals in Slitherlink, except the things that are diagonal in Slitherlink become horizontal/vertical in Slalom. For example, if you have two 3's horizontally or vertically adjacent in a Slalom puzzle, they must each have exactly one segment coming out amongst the two boxes that overlap both clues, and each has two segments pointing toward the outside. This is the equivalent of the diagonal 3's rule in Slitherlink. As with Slitherlink, this generalizes to the situation when there are 2's in a row/column between the 3's.

Another similar example would be two 1's horizontally or vertically adjacent, and neither along an edge. In this case again each will have one segment pointing out between the two 1's, but all the "outside" segments must be away from the 1's. This ends up looking sort of like "angle brackets" around the 1's.

More generally, you can often count number of segments working across a row/column of clues, in much the same way you can count along diagonals in slitherlink. So if your clues went (in a row) 3 3 2 1 2, you could tell how many segments came out of each clue to the left and right. Using (m n) to break a clue into m pointing left and n pointing right, it would be (2 1) (1 2) (0 2) (0 1) (1 1) in this case.

Another important strategy has to do with the no loop constraint. This is, I think, a common source of trouble for solvers, with a lot of folks posting Fehlversuche up at Croco-Puzzle because they didn't realize their solution violated this constraint. One of the key result of the constraint that every single segment must connect to one of the walls. In particular, you can't have any completely self-contained little mini-graph. If anything, this aspect of strategy feels a bit like Hashi strategy. Visually it's often pretty obvious if you create an "island". Let's look at a couple common examples.

One easy situation that occurs pretty frequently is two 1's diagonally adjacent, with neither along an edge. In this case, the segment between the two 1's must point so that it does NOT connect the 1's. If the 1's were connected, they would both be complete, with no additional segments coming out from them. That means their segment would never connect to the edge (it ends up completely surrounded by a loop). Note that if one of the 1's does touch an edge, it's perfectly fine for the diagonal segment to connect them. Similarly, if you have a 2 with two different 1's diagonally adjacent to it, and nothing on an edge, you can't connect both of the 1's to the 2.

Of course you must also avoid obvious small loops, like closed squares.

[detuned]

So you can generalize that observation about diagonal 1 clues to avoid situations where you have closed loops... sometimes it can sometimes be very hard to identify these, especially with larger grids. However, these closed loops happen if and only if there is at least one slalom "path" which is not connected to the edge of the grid. This comes in useful in particular where you have an internal slalom path without much branching - the point is you want to avoid connecting these to 1's so that these slalom paths have an opportunity to connect to the edge.

[detuned]

I should add that most of my own problems with this puzzle type, at least on croco, occur from misclicking things without realising...

[motris]

sknight has offered up all the common patterns I reboot in my mind before starting a croco-slalom. The second level thinking that is good to keep fluid in mind is looking at growing board situations for shut-downs. Yesterday's had a really large region bounded by just a 3. If two opposite diagonal segments came out from it, that would shut off the region entirely. Spotting this quickly let me get a top 10 time. It is the best deduction I've ever seen left behind in a slalom puzzle from any source.

I will say when solving on paper I turn the paper 45 degrees as I find tracing the paths easier when vertical and horizontal than otherwise. I don't know if this is just me but I almost always turn diagonal puzzles into up-down-left-right puzzles if I can.

On Tom's point, my most common error on croco's Slaloms (2 or 3 times I think) is leaving a cell blank, probably from checking too fast and the lines being thin. My one double Fehlversuche was from closing off a region, and also leaving one blank. That was particularly stupid.

[furudo.erika]

That's good advice in the previous couple of posts that I wish I'd read before dropping myself off the UK croco-ladder yesterday by mis-clicking and leaving a empty square

I really like Hall of Mirrors as a puzzle name, but just about anything would be better than the current name so I can stop getting it mixed up with Suraromu

[Mephisto]

Some words to Suraromu (if ypou are interested): Suraromu is not really an Japanese word, it's an English Word written in Hiragana/Katagana in phonetic spelling and then konverted to Romaji (Japanese in roman letters):

スラローム -> Suraromu -> S[u](r|l)a(r|l)om[u] -> Slalom

In Japanese there is no difference between R and L; in Romaji always R is used (even if L is correct). The Us are inserted for formal reasons only.

~ÔttÔ~

[Nix]

I thought it would be a good idea to write a bit about the deductions for Croco-Puzzle's only puzzle type in which I have consistent top results and with which many seem to struggle. It has taken me a huge number of computer-generated puzzles to get where I am, and I still remember how hard it was to first get a grip. There's something quite unintuitive about all of it. I've learned the rules simply by staring at puzzles long enough trying simple what-ifs mostly in my head until something ends up in a quick contradiction, then trying to figure out why and simplifying the setup as far as possible without losing the result. It has been an enjoyable process but I can imagine that one may want to skip some of it.

It would be much better to illustrate the patterns and deductions graphically but I don't have the time. I don't spell them out completely either, so you may have to experiment if you expectedly can't just see the deduction from the setup. I also don't really go into any more rarely needed tricks. A lot has already been said in this thread, but maybe it will benefit someone to read it again in different words.

---

Seeing when a loop is nearly formed and must be broken is much a learned skill and not much can be said about it. I do it by looking for shapes with only one connection to suitable blank (unsolved) cells and no existing connection to an edge, rather than trying to see the surrounding near-loop itself. One thing to notice is 1-clues or functionally equivalent shapes behind the blank cells, since connecting to the 1 doesn't help reaching an edge, and so another way must be found.

The only common thing for which you want to concern yourself with diagonally neighboring clues, is pairs of 1-clues and/or larger surrounded shapes that can only have one more line out, e.g. a 2-clue already connected to a 1-clue not on the edge.

Most of the deductions are in pairs of horizontally or vertically neighboring clues, possibly with 2-clues in between. A very useful rule is that adding any number of 2-clues between any pair of clues that yields a deduction, still yields the same deduction in the ends.

Most often you can deduce that the lines in the two cells between two clues must be in the same direction, giving exactly 1 connection to both clues whichever way the lines (both) end up being. Simple case: 1-clue on the edge, with another clue next to it in the middle. This can either fully determine the pair of cells behind a clue or require them to also have mutually parallel lines. This type of thinking gives the above rule of added 2-clues. Arrangements with unintuitively little known information that still lead to this include "1-1" and "3-3" always, "2-2" with several different arrangements of one given line behind each of the 2s, "1-2" with a line connected to the 2 behind it, and "3-2" with a line not connected to the 2 behind it. All the mentioned patterns give all of the cells behind the clues, and the fact that the cells between them are identical either way.

Another common horizontal or vertical deduction is a \/ formed by given lines with a 3-clue one step above it. The 3 can't have two lines coming down from it (closing the \/) so it must have two above it. Almost the same applies if the 3 is replaced with a 2 with a line not connected to it above.

Knowing these rules and reliably noticing the patterns when solving are two different things, and for most solvers I can bet that most often when stuck you have some such relatively simple deduction available and have just missed it. Of course you also need to see the trivial patterns where a clue needs all blank neighbors to connect to it, or already has all the needed connections. Also the simple U-shaped dead-end that requires a line to not close it.

There are more interesting deductions concerning possible paths for a shape to connect outside (impossible "around" a 3 for example), forming a "dead-end U" next to a 1, clues in the open with only one diagonal neighbor with more than one available connection, a clue with a pair of lines that together would close a loop (like the 3 above a \/), many sets of known-to-be identical cells interacting, and others, but I'd have trouble explaining them with even the readability of the above. I might return with pictures at some point.

[kiwijam]

Amazing! Many many thanks to Nix for this tutorial.

For those of you unaware, it is fair to say that Nix is "by far" the best in the world at croco-Slalom puzzles http://www.croco-puzzle.com/Rating/rating.php?typ=slalom. In fact his Slalom-rating is so high that it is higher than anyone else's rating in any puzzle http://www.croco-puzzle.com/Rating/ewigetabelle.php.

I'd love to see your advanced deductions if you have time. Any example diagrams wouldn't need an explanation ("a picture speaks a thousand words"), and it would be very interesting to see what 'patterns' you look for when everyone else gets stuck and starts guessing.

Also I'm curious, do you use a mouse, or keyboard, or both for solving Slaloms?

[Nix]

You're all very welcome... I just hope this doesn't kill my rating in Slalom, although now is not a bad time since I already dropped many perfect results away from the record 2988 by losing by 2 seconds to Kosi (pretty much the one other very good slalomist in Croco-Puzzle, no offence to anyone) in one quick puzzle.

I do hope I find the time and the enthusiasm to post diagrams for the more advanced deductions at some point, but no guarantees. Now if you're hoping for even more advanced techniques than those mentioned, I'm not sure if I know any. I did already list all I remembered (counting some variations), just thinking that there must be some I forgot.

Also I'm curious, do you use a mouse, or keyboard, or both for solving Slaloms?

Oh, there's absolutely exactly one way to do it right (well for me there is) – using the mouse to point and Q and W (and space) to mark. If right-clicking produced the other slant and reclicking with the same button did nothing, full-mouse + space would be sensible and maybe even faster. As it is, it's a nightmare to double-click even regularly, never mind using the mouse for correcting misclicks.

[transkrautor]

If you are not afraid of a few German words you can also find puzzles for beginners in the beginner's section of the puzzle portal at logic-masters.de. If you search for the word Slalom you will find three puzzles called 'Slalom für Anfänger' (which means 'slalom for beginners' in English). These will illustrate some of the techniques that Nix wrote about. And don't worry about the German headings: the slalom puzzles for beginners have both German and English text, it is just that the puzzle portal does not allow alternate language variants for headings.

And thanks a lot for sharing your techniques, Nix!

[PuzzleScot]

Ooh, I like that (beginners area)! Thanks for posting

Here's a 'trick' I use, that I haven't seen mentioned yet: Observing that all line segments form paths, every path must reach the edge of the grid, otherwise it will become an island within a closed loop. eg, (except at the edge of the grid) if you have a 1 connected to a 2, the other line from that 2 cannot go to another 1. Likewise, 2 diagonally adjacent 1s can never be connected. [edit: oops - I see it's mentioned in part 3 of the slalom guide - oh well]

The general rule I use with 3s is "if both of these cells have lines emanating from the 3, does that cause a problem?" - If so, then the other 2 cells round the 3 must contain lines emanating from the 3. (I deliberately choose 2 cells that are likely to cause a problem, so I can fill the other 2 in as definites)

[Nix]

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.

[Reveal] Spoiler: diagrams

slalom-1.png (3.27 KiB) Viewed 2445 times

slalom-2.png (1.66 KiB) Viewed 2445 times

slalom-3.png (2.54 KiB) Viewed 2445 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!

[Reveal] Spoiler: Solution to diagram 1

slalom-1.png (3.27 KiB) Viewed 2445 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}.

1. 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.
2. Similarly, the top 1-clue gives TL=TR.
3. 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.
4. 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.

[Reveal] Spoiler: Solution to diagram 2

slalom-2.png (1.66 KiB) Viewed 2445 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 "\".

[Reveal] Spoiler: Solution to diagram 3

slalom-3.png (2.54 KiB) Viewed 2445 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.

1. 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.
2. 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.
3. 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.

beginner's section of the puzzle portal at logic-masters.de

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.

[driv4r]

Tried to solve today's Slalom using all the instructions & tips given above but still ran into problems and had to guess/try to get it solved plus got it wrong as well, so had to find a way to remove/add lines. Added to that a fehlversuch meant a time like 1hr 31min 21sec.

Tbh, I only understand one of the tips Nix has given above: 'Another common horizontal or vertical deduction is a \/ formed by given lines with a 3-clue one step above it. The 3 can't have two lines coming down from it (closing the \/) so it must have two above it. Almost the same applies if the 3 is replaced with a 2 with a line not connected to it above.'

Others are not clear at all And I've no idea what those 3 images are about. I'm not here on the forum to solve puzzles but to learn how to solve and to solve I go to CrocoPuzzle!

Oh well, maybe you are just too pro...

I actually do understand most of the sknight's tips above but for example this one is unclear:

[---]
More generally, you can often count number of segments working across a row/column of clues, in much the same way you can count along diagonals in slitherlink. So if your clues went (in a row) 3 3 2 1 2, you could tell how many segments came out of each clue to the left and right. Using (m n) to break a clue into m pointing left and n pointing right, it would be (2 1) (1 2) (0 2) (0 1) (1 1) in this case.
[---]

To sum up this puzzle has annoyed me a lot once again, though this time I somehow got it solved in the end. Maybe it's just not for me
One positive to take as well from all that...I'm just glad I'm not in a contest this week, would be a definite point for my opponent.

[detuned]

The only observations i've ever needed for any of these puzzles are a few spot tricks with horizontally/vertically adjacent 1's and 3's, with diagonally adjacent 1's and a wary eye to make sure I'm not placing an edge which boxes in a 2 or 3 too much with chains of horizontally/vertically adjacent 2's and 3's - together with the key observation that any edge path in the grid must connect to the border. That said, you did need to watch your step with today's slalom puzzle.

[Nix]

Others are not clear at all And I've no idea what those 3 images are about. I'm not here on the forum to solve puzzles but to learn how to solve and to solve I go to CrocoPuzzle!

Sorry about that. There's a limit to how much can be expressed with words, and I'm not the best at it, especially not in English. I could have been more verbose too to leave less to figure out, but meh. Images for all the deductions would have been great, but most of them are already covered here pretty nicely:

beginner's section of the puzzle portal at logic-masters.de. If you search for the word Slalom you will find three puzzles called 'Slalom für Anfänger' (which means 'slalom for beginners' in English). These will illustrate some of the techniques that Nix wrote about. And don't worry about the German headings: the slalom puzzles for beginners have both German and English text, it is just that the puzzle portal does not allow alternate language variants for headings.

For the advanced techniques I pictured, I figure I will explain their solutions, but not right now.

Edit: But now. See the original message above.

[driv4r]

After struggling a lot with yesterday's Slalom and only getting it solved in the end thanks to bifurcating (using blue colour and +/- button/strufe) I have a couple of questions regarding the tips above.

Most of the deductions are in pairs of horizontally or vertically neighboring clues, possibly with 2-clues in between. A very useful rule is that adding any number of 2-clues between any pair of clues that yields a deduction, still yields the same deduction in the ends.

So, as I understand if for example there are 3 2 3 in a line next to each other then 2 lines would still go out of both 3s and one line would go in towards the 2? And same goes for 3 2 2 3, 3 2 2 2 3 etc.?

Most often you can deduce that the lines in the two cells between two clues must be in the same direction, giving exactly 1 connection to both clues whichever way the lines (both) end up being. Simple case: 1-clue on the edge, with another clue next to it in the middle. This can either fully determine the pair of cells behind a clue or require them to also have mutually parallel lines. This type of thinking gives the above rule of added 2-clues. Arrangements with unintuitively little known information that still lead to this include "1-1" and "3-3" always, "2-2" with several different arrangements of one given line behind each of the 2s, "1-2" with a line connected to the 2 behind it, and "3-2" with a line not connected to the 2 behind it. All the mentioned patterns give all of the cells behind the clues, and the fact that the cells between them are identical either way.

I just don't understand this part, could you explain it more clearly and in more detail?

[kiwijam]

So, as I understand if for example there are 3 2 3 in a line next to each other then 2 lines would still go out of both 3s and one line would go in towards the 2? And same goes for 3 2 2 3, 3 2 2 2 3 etc.?

Yes, you don't know where the lines are around the 2s, but outside each 3 you can draw 2 lines. e.g. >323<, >3223<, >32223<.

Most often you can deduce that the lines in the two cells between two clues must be in the same direction, giving exactly 1 connection to both clues whichever way the lines (both) end up being. Simple case: 1-clue on the edge, with another clue next to it in the middle. This can either fully determine the pair of cells behind a clue or require them to also have mutually parallel lines. This type of thinking gives the above rule of added 2-clues. Arrangements with unintuitively little known information that still lead to this include "1-1" and "3-3" always, "2-2" with several different arrangements of one given line behind each of the 2s, "1-2" with a line connected to the 2 behind it, and "3-2" with a line not connected to the 2 behind it. All the mentioned patterns give all of the cells behind the clues, and the fact that the cells between them are identical either way.

As mentioned above, look at the nice pictures at http://www.logic-masters.de/Raetselport ... ?id=0000XH . All Nix's situations are shown there.

[PuzzleScot]

Most often you can deduce that the lines in the two cells between two clues must be in the same direction, giving exactly 1 connection to both clues whichever way the lines (both) end up being.

I just don't understand this part, could you explain it more clearly and in more detail?

In other words: Sometimes, based on other information nearby, on one side of a clue number, there must be 1 line going to the clue, and one not. (ie, they are parallel).

[apollo1001]

Oooh, fabulous thread guys! I was deducing many of these facts yesterday from scratch in the puzzle, and wondered if we had a central set of 'standards' to look out for. I'll remember to search the forum in future before attempting any puzzle type!