Solve five Slitherlinks, each with a unique solution.

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

• ... five? – Rand al'Thor Mar 28 '17 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? – ais523 Mar 28 '17 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. – Chris Mar 29 '17 at 11:01
• @EricDuminil I've updated the answer a bit to make this clearer. – sTertooy Mar 29 '17 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 :) – Sconibulus Mar 28 '17 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. – GoldenGremlin Mar 28 '17 at 15:34
• I edited my question to make it clear that solutions should be unique. – GoldenGremlin Mar 28 '17 at 15:36
• @KaranAtree This is unsolvable without tetrominoes. Your second 2 on the left of fifth has only one adjacent line. – Techidiot Mar 28 '17 at 16:18
• @Silenus Congratulations, your comment has inspired a new question :-) – Rand al'Thor Mar 28 '17 at 16:43