Notation

Across Skyscraping, I sometimes use a textual notation for lanes of a Skyscrapers puzzle. Since it is’t always practical to create a full diagram like this…

…for every situation, this more efficient notation allows us to more efficiently discuss possibilities and deductions.

This page describes the syntax of this notation. It’s hopefully pretty self-explanatory!

Solved Cells

Take the following lane:

We notate this as:

$$ext{5 | 1 2 3 4 5 | 1}$$

The vertical pipes “$|$” denote the edges of the grid, separating the skyscrapers from the clues.

Keep in mind, the direction of the ‘notated’ lane doesn’t need to be the same as in the original grid. If we were interested in looking right-to-left in the original lane, we would instead write it as:

$$ext{1 | 5 4 3 2 1 | 5}$$

For consistency, the direction of interest will always be left-to-right. So when you see $p \text{ | 1 2 3 ... | }q$, that means we’re focusing on the $p$-clue.

The direction of the original lane is irrelevant – it very well could be a column!

Here, $\text{1 | 4 1 3 2 | 3}$ means we’re looking up the column, while $\text{3 | 2 3 1 4 | 1}$ means we’re looking down it.

Pencilmarks

For a lane with pencilmarks:

We notate this as:

$$ext{4 | [12] [23] [34] 5 [1234] | 2}$$

We surround the candidates with square brackets “$[]$”.

Unsolved Cells

Too many pencilmarks gets a bit unwieldy, though. Sometimes, we may just want to ignore the candidates, if they’re not relevant to our current focus.

In this lane, we’re focusing on the $3$-clue. What goes between the $5$ and $4$ isn’t currently of interest to us.

We notate this as:

$$ext{3 | [12] [23] 5 _ 4 | 2}$$

We use underscore “$\_$” to denote an unsolved cell without explicit candidates.

We’ll use this a lot in large puzzles where explicitly showing the candidates would otherwise be more distracting than helpful.

If things become especially dire, we might omit a whole bunch of cells at once with ellipsis “$...$”. For the above lane, this would give $\text{3 | 4 [56] 7 ... 3 | 3}$.

All this helps keep the notation clean and focused!

Half-Lane

If we don’t care what’s beyond the lane peak at all – though this is rare, since usually we need to consider the whole lane to perform deductions – we can just omit the rest:

$$ext{3 | [12] [23] 5}$$

Sets

We often need to talk about a particular set of numbers.

In the above lane, we say we have used $\{35\}$, and have not used $\{124\}$.

We surround the numbers of interest in curly braces “$\{\}$”, to represent a set where order is irrelevant (we’ll usually sort it for clarity).

Peaks

Sometimes, we may talk about a specific chain of skyscrapers in a half-lane. Order does matter here, since we’re referring to the exact order the skyscrapers come in the puzzle.

In the above lane, we say that we have placed $(24153)$, and can see $(245)$.

We surround the numbers of interest in rounded parentheses “$()$”, to represent an ordered list of skyscrapers.

$()$ isn’t really ideal, but it’s kinda the best we’ve got since $[]$ and $\{\}$ are now taken lmao.

Notes

Throughout all the notation we omit commas “$,$” to keep the notation compact and efficient. Since we never look at Skyscrapers of double-digit sizes, we can safely assume each digit is one individual skyscraper.