Stuck on 6x6 skyscraper puzzle

I don't know how to continue. Can anyone tell me what is the next number I can deduct? Here is the puzzle. The numbers along the edges are clues, and the second grid is the possibilities. Feel free to tell me if I made mistake. If possible show me a few further steps, or some general techniques I can use to determine next number.

Transcription of current puzzle state, stolen from @Rand al'Thor's answer:

    3 2 2 3 2 1

= = = = = =
4  | | | | |5|6|  1
= = = = = =
2  | | | | |4|5|  2
= = = = = =
2  | | | |5| | |  3
= = = = = =
1  |6|5| | |3|4|  3
= = = = = =
2  |5| | |6| | |  2
= = = = = =
3  | | |5| |6|1|  2
= = = = = =

3 4 2 2 1 5


Firstly, just to put your puzzle in editable format:

    3 2 2 3 2 1

= = = = = =
4  | | | | |5|6|  1
= = = = = =
2  | | | | |4|5|  2
= = = = = =
2  | | | |5| | |  3
= = = = = =
1  |6|5| | |3|4|  3
= = = = = =
2  |5| | |6| | |  2
= = = = = =
3  | | |5| |6|1|  2
= = = = = =

3 4 2 2 1 5


Now consider the square in the top row, fourth from the left.

• Let's assume it's 4. Then ...
We see four skyscrapers from the left of the top row, so the top left corner must be 3.
We see three skyscrapers from the top of the left column, so 4 must be between 3 and 6 in this column, so the bottom left square must be 1 or 2.
We see three skyscrapers from the left of the bottom row, so the bottom left square is taller than its neighbour to the right. So the bottom left square must be 2 and its neighbour 1. But we already have a 1 in this row, contradiction.

 3 2 2 3 2 1

 = = = = = =
4 |3| | |4|5|6| 1
 = = = = = =
2 | | | | |4|5| 2
 = = = = = =
2 | | | |5| | | 3
 = = = = = =
1 |6|5| | |3|4| 3
 = = = = = =
2 |5| | |6| | | 2
 = = = = = =
3 |2|1|5| |6|1| 2
 = = = = = =

 3 4 2 2 1 5

• Let's assume it's 2. Then ...
We see three skyscrapers from the top of that column, so the square just below must be 1.
But now there are no options for the fourth square of the fourth row: it can't be 1, 2, 3, 4, 5, or 6 by Sudoku rules on the row and column. Contradiction.

 3 2 2 3 2 1

 = = = = = =
4 | | | |2|5|6| 1
 = = = = = =
2 | | | |1|4|5| 2
 = = = = = =
2 | | | |5| | | 3
 = = = = = =
1 |6|5| |X|3|4| 3
 = = = = = =
2 |5| | |6| | | 2
 = = = = = =
3 | | |5| |6|1| 2
 = = = = = =

 3 4 2 2 1 5

• You've already reduced the options to 2, 3, 4, so now you know that square must be 3.

Now, in the fourth column,

there's only one possible place left for 4, namely the bottom row. (The other blank spaces in the column are in the same row as existing 4's.)

Then, in the bottom row,

the only numbers left are 2 and 3; we see three skyscrapers from the left, so it must be 3 then 2.

You can probably do the rest yourself, after getting past that one tricky step.

• Great explanation, problem solved, thank you! Commented Feb 16, 2019 at 15:01
• @victory Happy to help! These puzzles are fun :-D Commented Feb 16, 2019 at 15:01
• I do find that sky scraper needs the ability to undo. In Simon Tatham's version I enter a guess and delete (not undo) and repeat a couple of times. Then when you undo that number will flash a couple of times. You can even have nested guesses that way and keep track over several sessions Commented Feb 16, 2019 at 15:41