Solve five Slitherlinks, each with a unique solution.

Please say a word regarding (or show an image depicting) each solution method.

• ... five? Commented Mar 28, 2017 at 15:09

The four slitherlinks give tetrominoes as answers. You can combine these to make a new grid (bottom left), which again has a unique solution (bottom right).

• Ugh, you beat me by about 20 seconds. Do I have to post the answer unexplained first, and then edit in an explanation, like I do in PPCG? Commented Mar 28, 2017 at 16:01
• @EricDuminil: You may be confusing what the first image represents. The four given grids have solutions that are tetrominoes. They can be put into a four by four grid in only one way which is what the first image shows. The second image then shows the solution to that puzzle. Commented Mar 29, 2017 at 11:01
• @EricDuminil I've updated the answer a bit to make this clearer. Commented Mar 29, 2017 at 11:19

I looked at other people's answers for the first four grids, and got the solutions from those. However, here's what I think the fifth grid is (minor spoilers for the other four grids):

Place the four tetronimoes from the solutions to the first four grids together into a 2×2 square, without rotating them. There's only one way to do it:
+-+-----+
|?|? 2 ?|
| +-+-+ |
|1 3|3|?|
| +-+ +-+
|?|?|2 ?|
+-+ +-+ |
|3 ? ?|?|
+-----+-+

It has a unique solution:

..+-+-+-+
.?|?.2.?|
..+.+-+.+
.1|3|3|?|
..+-+.+.+
.?.?.2|?|
+-+-+-+.+
|3.?.?.?|
+-+-+-+-+
This is copied directly from my notes; I was using + for an intersection the line definitely went through, and . for an intersection or edge it definitely didn't go through. The main observation is that the only way to get a line through a pair of 3s is to zigzag it past; if you zigzag it the other way, the 2 forces it to get stuck in the top-left corner, so the only possible path for the line goes round like this.

I've found four slitherlink solutions, but I'm not certain how to construct the fifth...

I initially thought the items in the tetrominoes were potential values for that square in a 2x2 Slitherlink, but there's no way to make that work without rearranging the squares. (ex. 3/3 on the right side is a contradiction, and 3/1 would force a 1 at the bottom left, which only has options for 3 or 2.)

I think I have got the Answer to all 5:-

• ...well now I feel dumb :) Commented Mar 28, 2017 at 15:27
• Nice try, but your fifth grid does not yield a unique solution and is thus not the intended Slitherlink puzzle. There are very many solutions to your fifth grid. The intended fifth grid has a unique solution. Commented Mar 28, 2017 at 15:34
• I edited my question to make it clear that solutions should be unique. Commented Mar 28, 2017 at 15:36
• @KaranAtree This is unsolvable without tetrominoes. Your second 2 on the left of fifth has only one adjacent line. Commented Mar 28, 2017 at 16:18
• @Silenus Congratulations, your comment has inspired a new question :-) Commented Mar 28, 2017 at 16:43