## Monday 5 December 2011

### Puzzle #69: Four Colour Loop

This puzzle is a unique mix of four different genres: Maxi Loop, Liar Loop, Double Back and Country Road. I haven't ever seen genres combined in this way before. I've combined them by using the Four Colour Theorem. This theorem states that when a plane is split up in different regions, you'd only need four different colours to colour all regions without having any regions of the same colour touching by a side.
I'd been thinking of using this for a while, but it was hard to find a good set of puzzle rules that would work in a single puzzle. When I finally had four rules I thought would work, it took a while to even get a solid opening that would get the puzzle going. It took me a long while to get a good puzzle. I had to edit the puzzle a bunch of times as I kept running into mistakes I had made while constructing. I've worked on this puzzle on and off for a few weeks. I hope you all enjoy it. I'm warning you that it isn't easy, but it is completely logically solvable. It will probably take a while to see how the rules interact and what restrictions there are to assign the right rules to the right regions.

Four Colour Loop

General rule:
Draw a single closed loop by connecting the centres of cells horizontally and vertically. The loop doesn't touch or cross itself. The grid is split up into different regions. Each region obeys one of four rules. No two regions with the same rule touch eachother by a side anywhere. They are allowed to touch by a corner.

Region rules:
The loop enters and exits the region once. The number in the region indicates the amount of cells the loop runs through. (Note: This means the loop doesn't run through all cells in the grid)

Double Back:
The loop runs through all cells in the region. The number in the region indicates the amount of the times the loop enters and exits the region.

Liar Loop:
The loop runs through all cells in the region. The number in the region indicates that the loop never runs through this many cells consecutively when traveling through that region. It is always more or less.

Maxi Loop:
The loop runs through all cells in the region. The number in the region indicates the highest amount of cells the loop runs through consecutively when running through this region.

 Click to enlarge

1. Oh s***.

Er, when you say "Each region obeys one of four rules," does that mean it won't obey any others? For example, several regions are labeled 1; if any obeys the Double Back rule, it will then obey the Liar Loop rule.

And of course, Country Road is inconsistent with everything, and Maxi and Liar Loops are inconsistent, but Double Back could be obeyed concurrent with one of the Loop rules.

It's so big....

2. This comment has been removed by the author.

3. The rules aren't mutually exclusive. So a Liar Loop region with a 3 could have 3 entries and exits.

The basic principle to the opening is finding a region that can only have a single rule and work on assigning rules to regions from there on. It's not a tricky deduction, it's just some work to find the right one. A lot of numbers will instantly be assignable to different rules, so you can dismiss the opening being there.

It's big and hard and not really a puzzle to solve quickly. But I think it will be enjoyable once you get started.

4. I figured that out minutes after posting, but I was going to bed.

A DB-exclusive region with the clue X would need at minimum (a) one length of X cells to violate LL, and (b) one length of more than X cells to violate ML. So since all regions here are hexominal(?), DB regions must contain clues of 2 and one run each of 2 and 4. I've found a DB region, and it cannot be split like that.

5. Total solving time: 111 minutes, but I watched episodes of series in the meantime. In the first 41 minutes I could solve just one quarter, maybe I was a bit tired after the midnight.
Thanks a lot, great puzzle.

An other comment, you wrote you dont want to post puzzles regularly. Is this changed? :)

6. A short quote from the Smallville s10e01:
"So much better in technicolor." :-)

7. Thanks a lot for this awesome puzzle. I enjoyed it a lot and I really love the concept.

It took me a couple of tries because I always messed up somewhere, but in the end I got through cleanly. There's one place where uniqueness can be applied usefully, though I avoided it on the last solve.

8. @ Valezius: That seems like a pretty respectable time in my opinon. It's not meant for speed solving.
If I have something I want to post, I'm going to post it. It just won't be a daily thing. Maybe a few times a week, who knows. I did the 24 Hour recap, just because I had them and figured I could just as well post them for everyone to see.
I actually drew up a nice looking coloured solution in case someone needed it.

@ Rob: Thanks. I think I know what place you mean that uniqueness could be useful. I couldn't find a way around it without making it easier. So I figured people who would want to use it would, and the rest would see at what point that option would be resolved in another way. My view on uniqueness is that I only use it during timed solves but not when I'm just solving for fun. The same goes for guessing.

9. I have about the bottom half of the regions and a third of the loop in pieces. I don't want to use uniqueness (like the CR at row 12, column 6). What blind alleys can I explore so I can rule them out?

10. That CR cage gets resolved only to the end. From what I assume you have you should focus on the right side of the grid. There's 1 tricky region to figure out on the right side in the middle, but after that you should be able to figure the loop over the right side upwards.

11. Done! I'd actually solved most of that 1, but your hint got me to focus on the 4 above.

12. Congrats. (The 1 wasn't meant to make you focus on the region with the 1. I was thinking you'd be at the point where you needed to figure out that 4 cage as that is probably the trickiest one to figure out.)