Case: Ascendant

The clue of $2$

In a $2$ -clue lane, the $N-1$ skyscraper must either be in the head cell or past-peak.


2				7


2	6			7	6	6	6

Recursion

If the $N-1$ skyscraper is past-peak, this case recurses with the $N-2$ skyscraper, $N-3$ , and so on.


2			N	…	N-1	…	N-k


2	*N-k-1*		N	…	N-1	…	N-k

Explanation

Really, this case arises because the $N-1$ skyscraper cannot be both pre-peak and not in the head cell (in a $2$ -clue lane).


2		5		6

If this is the case, the $N$ and $N-1$ are guaranteed to be visible. But we also have unsolved cells before the $N$ , of which at least 1 will be visible too. This means we’ll have at least 3 skyscrapers visible in total, which contradicts the $2$ -clue.


2		5		6

This cell before the $5$ -skyscraper ( $N-1$ ) must be shorter than both $5$ and $6$ , so must be visible. But now we have 3 visible skyscrapers, not 2 as required!

Therefore, if the $N-1$ is pre-peak, it has to go in the head cell.


2	5			6

The only pre-peak cell for the $5$ is the head cell so it obscures the other pre-peak skyscrapers.

If it’s past-peak, well, then it makes no difference to the half-lane where it goes, so it can be anywhere.


2				6	5


2				6		5

Both of these are possible, since we’re outside the half-lane so the $2$ -clue can’t affect us.

Recursion

Explanation

See Also