Case: Blockade

$N$ opposite $2$ -clue

In a $2$ -clue lane with the $N$ -skyscraper (lane peak) in the tail cell, the head cell must contain the $N-1$ skyscraper.


2			…		N


2	*N-1*		…		N

Recursion

This case recurses by incrementing the clue, and increasing the number of high skyscrapers in the back of the lane. This decrements the skyscraper in the head cell.



2				6
3			5	6
4		4	5	6
5	3	4	5	6



2	5				6
3	4			5	6
4	3		4	5	6
5	2	3	4	5	6

For more details see Recursion & Abstraction.

Examples

Case 1




2					5




2	4	5

Explanation

This is a super common pattern, and one of the first deductions you might learn!

Blocking the Bridge

Let’s take this lane in a 5x5 puzzle.


2					5

The $2$ -clue means we need 2 skyscrapers visible. 1 of those must be the lane peak (which is always visible).


2					5

The $5$ is always visible since the other skyscrapers $\{1234\}$ are all shorter than it.

Then, let’s consider where the other skyscrapers can go. We’ll start with the $4$ -skyscraper. Let’s try placing it here:


2			4		5

Now we’ve guaranteed we can see at least 2 skyscrapers. But, we should continue our hypothetical deductions, because we might also see more than 2 skyscrapers, which would contradict the $2$ -clue.

Indeed, we could now place the $3$ in the head cell, which would mean we now see 3 skyscrapers, not 2!


2	3		4		5

Oh no, contradiction! In the lane, $(345)$ will be visible, so the $2$ -clue is not satisfied.

The same argument applies for all the other cells – except the head cell!


2	4				5

Here, the $3$ can no longer be placed before the $4$ , because $4$ is the very first skyscraper. So now we only see $(45)$ – 2 skyscrapers, as required.

You can check that no matter how we place the rest of the skyscrapers, only these 2 will be visible, since the $4$ will hide all of $\{123\}$ .

Beware the Prerequisites

If you’re just starting out, be careful! This deduction only applies if the lane peak is in the tail cell (the last cell).


2				5

When the lane peak is not in the tail cell, the $N-1$ could very well be hiding behind it.


2				5	4

We can’t be certain the $4$ is in the head cell, because it could be hiding here behind the $5$ .

More precisely, it only applies when the lane peak is effectively in the tail cell – see Case: Outflanked Blockade.