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This puzzle is part of the Monthly Topic Challenge #13: Variety Slitherlinks.


A watermill is going to be built next to a cozy town. You are the designer of the aforementioned mill/slitherlink, and you have been given some requirements for building.

  • When the mill is operational, every cell not inside the slitherlink will be filled with water. The water acts under the force of gravity: it will always attempt to move downwards. If it cannot move downwards, it may move randomly either left or right.

  • There are certain water reservoirs that the water must drain to, shown as cages.

  • Every water tile must uniquely drain to one reservoir. However, there is one special case where this does not apply.

  • In the bottom, two reservoirs have overlapping drainage areas. The purple tile of water intersects the reservoir 1 down, 2 left from it and the reservoir 4 down, 2 right from it.

  • Normal Slitherlink rules apply.


slitherlink

Solve on Penpa (answer checking not enabled because I legitimately can't figure out how).

Author note: there's a flaw with the Penpa link where apparently anyone can edit and it'll change the board.

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    $\begingroup$ What's up with the outer border of the grid? Why is it solid? And what about the blue line near the bottom - is that just a given part of the loop? $\endgroup$
    – Deusovi
    Aug 25, 2023 at 23:31
  • $\begingroup$ @Deusovi Yes. that's a given part - I figured it would fit with the theme. The outer border is solid because of a style preference. Nothing enigmatic hidden in here. $\endgroup$ Aug 25, 2023 at 23:51
  • $\begingroup$ If the problem with Penpa is affecting you, you can just clone the link - it should work from there. $\endgroup$ Aug 26, 2023 at 12:07

1 Answer 1

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Some opening deductions:

grid image 1 Here, I've used only basic Slitherlink deductions, and also blocked off the bottom of each reservoir.

Continuing,

I use the 3 in the lower left - it can only open up or right, and it can't open upwards or it would create a "pit" that water couldn't escape.
grid image 2
Now the reservoir in R7C2 can't continue right - if it did, it would spill downwards due to the 0. So we can continue the coloring arguments...
grid image 3

Now, noting the condition of the purple tile,

we can say that the 0 in the bottom right must have water.

More coloring along with the Slitherlink clues gets us this far. (This seems to contradict the rules for the purple tile - surely the water in it can go down so it can't go left. But the phrasing there is somewhat unclear.)

grid image 4

In the top left, we can use an interesting fact about 2 clues:

specifically, that the four cells around them must be split 50/50 between inside and outside the loop, regardless of the actual color of the clue itself. This lets us color several more cells...
top right section
...and then connectivity of the green section resolves some ambiguities.
grid image 5

Now, all that's left is the top-left corner.

The 1 in the top left must have an edge either above it or to the right.

grid image 6
Now consider the 2 in R2C2; it must either take both of the top and right edges, or neither of them. Either way, it connects to the dot in its lower right, so we can resolve the 3s...

...and from that, the rest of the solution falls into place:

finished grid

(Thanks to fljx and quarague for correcting a mistake in the first version of this answer.)

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  • $\begingroup$ Welp! I accidentally forgot to cross off some checks I made. I'll fix that link later. $\endgroup$ Aug 26, 2023 at 1:56
  • $\begingroup$ You can avoid bifurcation in the top left by ROT13(Gur ybbc ng gur gbc bs gur tevq zhfg tb nybat gur gbc be evtug bs gur 1, fb vgf obggbz naq yrsg rqtrf ner obgu abg ybbc. Gung zrnaf gur 2 orybj-yrsg vf fvggvat va n pbeare naq zhfg unir ybbc ragrevat/rkvgvat guebhtu gur obggbz-evtug naq gbc-yrsg pbearef. Naq gung zrnaf gur yrsg rqtr bs gur hccre 3 pnaabg or ybbc.) $\endgroup$
    – fljx
    Aug 26, 2023 at 8:20
  • $\begingroup$ Your 1 in the top right has 2 lines next to it, something is off around there. You can avoid the case-bashing around there with slitherlink deductions that hopefully lead to the one correct solution. First the 1 has a path feeding into it on top of 3, so it's one line is either top or right. Excluding bottom and left for this 1 implies that the 2 has either top and right or bottom and left. This implies that the to 3s on top of each other are connected like the number 2 (and not its mirror image). $\endgroup$
    – quarague
    Aug 26, 2023 at 11:39
  • $\begingroup$ Yep, I'm accepting this answer because it's essentially correct, but as @quarague mentioned, that 1 is slightly off... About the resovoirs and the purple tile, I was more thinking about the 'resovoirs occupy the same space' in theory where there wasn't any gravity and there were just regions. $\endgroup$ Aug 26, 2023 at 12:08

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