# Next step/strategies for an Area 51 (Slitherlink variant) puzzle

I've been doing quite a few Slitherlinks recently (and puzzles containing Slitherlink elements, such as Area 51). I'm familiar with all the patterns described in the Wiki article but I feel like there's some stuff I'm still missing with them.

I'm currently stuck on this Area 51 puzzle and would like some tips on what the next step(s) are.

The rules of Area 51:

Transcription:

You are building a labyrinthian fence enclosing Area 51. The fence must separate the aliens, kept inside, from the cacti, which are outside. The finished fence must make an enclosed circuit without touching or crossing itself. A cell containing an uncircled digit must be surrounded by that many pieces of fence. Numbers inside dotted-circles must be inside the fence and show how many cells can be seen up, down, left, and right (plus itself) from that cell before reaching a fence. On darkened circles, the fence must have a 90 degree turn with straight sections before and after. On white circles, the fence must continue straight with a turn before and/or after the circle. Thanks to David Millar who created this puzzle variety.

Also, if you're aware of any more advanced strategies, please mention anything relevant - I'm not sure where I'm falling down here.

Here's my current board-state:

• This is Krazydad. Just wanted to mention that although my Area51 puzzles are computer generated, my software is written in such a way as to guarantee the Area51 puzzles are solvable using logic, and don’t require trial and error. Deusovi’s solution looks good! May 6, 2021 at 13:58

Krazydad's puzzles are computer-generated, so there isn't necessarily a "nice" human-doable path through them. (In my experience, there hasn't been a puzzle without a reasonable path through it, but I've only done a couple dozen of them at most.)

There is a nice way to make progress here, though: take a look at the 1 closest to the bottom right corner.

You can rule out some options for it based on how it interacts with another clue.

Specifically, how it interacts with the 9 in its row.

If you use either the top or bottom edge, what happens?

And a giveaway:

If the 1 doesn't use either the edge to its left or right, then the remaining two cells in that row are both inside the loop. So then the 9 clue can see ten cells, and we have a contradiction. To prevent this, the 1 must use either its left or right edge.

Deusovi pretty much has the right answer, so this answer is just to elaborate a little on this in case you might not know about this technique (and also show the corresponding deductions you can make):

A key observation is that if a cell is to be located outside of a loop, there must be an even number of link segments between it and the border of the puzzle in all directions. If the cell is to be located inside a loop, then there must be an odd number of link segments between it and the border of the puzzle in all directions.

If the '1' in the bottom right corner were to use its top or bottom edge, then there cannot be any more vertical segments in the row the circled '9' because it will lead to the circled 9 seeing an even number of segments to its right and there is no more space to place another vertical segment somewhere to the right of the circled '9' (and you will violate the constraints of the '1' if you do).

So, there cannot be any more vertical segments for that row and instead you will get the following figure (assuming you placed a link to the top of the '1'):

Here, you have another contradiction because now the '9' sees 10 squares to its left and right. So, since placing a bottom or top link will eventually run into a contradiction, we can deduce that the link must go to the left or right of the '1'.

We can in fact go a little further with that deduction and using the key observation I made earlier.

We know that a link segment cannot be to the right of the '1' since it will violate either the '1' or the '2' in the bottom-right corner. So, it must go to the left of the '1'.

There are currently 2 link segments to the right of the circled '9', but we know there must be an odd number of link segments to the right of the circled '9'. So, there must be another vertical segment in that row and this can only be left to the link segment we just placed. This gives us the following figure:

Based on this, we can now make another deduction! Observe that now the circled '9' only sees 7 squares horizontally. This means it must see 2 squares vertically. The maximum it can see upwards is 1 since it will get blocked off by the '3'. So, it must see 1 square to its top and 1 to its bottom. Using this information and inserting the links correctly for connectivity will get us to:

And we have made significant progress on the bottom of the grid! The rest of the puzzle can be solved in a fairly straightforward manner, so I will leave that to you.