Case: Outflanked

In some cases which require the lane peak to be in the tail cell, we may still be able to apply the deduction if there are suitable constraints on the past-peak cells.

Outflanked Blockade

The cleanest example of this is Blockade. Usually, this is only applicable with the lane peak in the tail cell like so:

However, could we also apply Blockade in this situation?

Well, the lane peak isn’t in the tail cell, but there aren’t any unsolved cells past-peak. So all the places the $4$ can go are pre-peak.

And of course, the only valid one is the head cell.

Well, that just felt like Blockade, didn’t it?!

This works because the solved past-peak cells are irrelevant when considering the $4$. As far as the $4$ is concerned, the lane peak is in the tail cell, since the $4$ can’t be past-peak.

We can ‘outflank’ Blockade with other constraints, too! Here, we do have an unsolved cell past-peak:

We know from Successor that the $4$-skyscraper must either be in the head cell or past-peak.

But, notice the other $4$ already in the column of the tail cell. This means we can’t place a $4$ in the tail cell, since the two would conflict.

This means it must be pre-peak, and of course the only pre-peak cell it can go in is the head cell.

Again, this feels just like Blockade, right?

The other $4$ is cutting off that past-peak tail cell, so it’s like it doesn’t even exist. So really, the ‘lane’ as it appears to the $4$ we want to place is just these first 4 cells:

Which means, as far as the $4$ is concerned, the lane peak is in the tail cell, so Blockade does apply. Pretty cool, right?

We can perform this outflanking as long as all the past-peak cells have constraints preventing the $N-1$ skyscraper from being placed in them.

We’ve seen two such constraints here – a literally solved cell and a conflicting solved cell – but any others work also work. Two easily spottable concrete ones are:

• A laser:

• A couple:

Other more subtle ones will be revealed through pencilmarking ;)

Recursive Outflanked Blockade

Now take a look at this situation, and again we’ll try to apply Blockade.

Blockade usually talks about the $N-1$ skyscraper (here, the $5$), except here the $5$ has already been used in the tail cell.

But, just like before, since all the past-peak cells are solved, the lane peak is effectively in the tail cell. The structure we need is the same:

To obscure the intermediate cells, we need the tallest currently available skyscraper. Usually that’s $N-1$, but here $5$ has been taken, so the next tallest is $4$.

The logic is the same as Blockade, but we used the $N-2$ skyscraper instead of the $N-1$ skyscraper.

And we can keep going – even for a monstrous 9x9 puzzle, this recursive logic still holds.

Remember that the head cell skyscraper we want is the tallest available one, not just any available one. Here, the past-peak skyscrapers aren’t a consecutive set:

There’s $\{653\}$, with a gap of $4$ – that’s the skyscraper we want, not the ‘next’ ($2$).

Also, if even one past-peak cell is unsolved, this won’t necessarily apply!

The key constraint is that the lane peak must be effectively in the tail cell, which allows Blockade to still hold.

Recursive Outflanked Middle Ground

TODO