Glossary

This page defines all of the technical quasi-jargon I use throughout Skyscraping.

It really isn’t my intention to gatekeep beginners! In fact, I’ve made this partially for myself too, because it ensures:

• Consistency – one name for one concept.
• Unambiguity – well-defined concepts.
• Concise – no verbose “looking across this lane until we reach the skyscraper of tallest height”, just “looking across the half-lane”.

The first two hold me accountable!

I wouldn’t recommend reading through this entire page – instead, use it as a reference for when you come across non-obvious unfamiliar terms. The first usage of a keyword on every page will link to this page for convenient access.

backtrack

Since we spotted the contradiction early, luckily we can still backtrack.

To undo deductions.

candidates

$1$ and $2$ are the only candidates for this cell.

The numbers in pencilmarks, indicating which skyscrapers could be placed in that cell.

cell

A single square in the puzzle grid.

A cell is either:

• solved, containing one skyscraper
• unsolved, containing many candidates, which may be indicated with pencilmarks

I’ll also often use “Cell” to refer to the skyscraper inside it (metonymy!^↗).

closed

Solving closed Skyscrapers is fairly trivial, but solving open Skyscrapers is something else entirely.

A closed Skyscrapers puzzle has clues along every lane, such that it’s ‘closed off’.

With the maximum number of clues available, these Skyscrapers tend to require more work than thought, and aren’t particularly fun.

Also called “dense” for historical reasons.

clue

This lane has a clue of $5$, so we know it must have a sequence.

A number outside the puzzle grid.

An $n$-clue indicates that in the lane it is adjacent to, there are precisely $n$ skyscrapers visible. For instance, here the clue is $3$:

$$ext{3 | 1 4 2 3 6 5 |}$$

concrete deduction

A deduction which we don’t intend to backtrack on.

If solving Skyscrapers ‘properly’, this means we are making the deduction with certainty – we can be confident the deduction is correct because it is based on logic.

contradiction

Oh dear, we’ve found a contradiction. That means we must’ve gone wrong somewhere…

A situation where a constraint cannot be satisfied (is contradicted).

For instance, if half-lane has a clue of $4$, but our working in its current state means 5 skyscrapers are guaranteed to be visible, then this is a contradiction. We’ve violated the $4$-clue, and there’s no way to repair it.

Reaching a contradiction when performing concrete deductions means you’ve made an error in your solution. Reaching a contradiction when performing hypothetical deductions is very common, and is a strategy for ruling out potential solutions as impossible.

counterexample

We’ve found a counterexample, so we know this cell can’t contain a $3$-skyscraper.

A potential configuration of candidates which leads to a contradiction, which disproves the possibility of that configuration.

This is usually relevant while performing hypothetical deductions.

couple

Main article: Technique: Couples

2 unsolved cells in the same lane lane which contain the same 2 candidates.

In the grid above, the $[34]$ cells are couples, as are the $[12]$ cells.

Also called “twins” and “pairs”.

deduction

Keep looking for deductions, even where you’ve already looked!

A logical inference based on the currently available information.

For instance, in a lane $\text{5 | \_ \_ \_ \_ \_ \_}$ we can deduce the solution msut be $\text{5 | 1 2 3 4 5}$ using the rules of Skyscrapers.

Solving a Skyscrapers puzzle is just repeatedly performing deductions! Cases covers very specific logical deductions; Techniques covers general techniques that aid performing deductions when solving Skyscrapers.

dense

Another name for a closed Skyscrapers.

half-lane

One ‘side’ of a lane – the cells from the head cell up to and including the lane peak, as well as the clue (if any) on that side.

The 2 half-lanes in the above lane are $\text{3 | [12] [23] 5}$ (from the left) and $\text{2 | 4 [123] 5}$ (from the right).

head cell

The head cell can’t be too tall or we won’t be able to satisfy the clue.

The first cell in a lane.

In this lane, the head cell is the one containing the pencilmarks $[123]$.

See also: tail cell

grid

Refers specifically to the cells of the puzzle, without the clues.

An NxN Skyscrapers puzzle has a grid with dimensions $N \times N$.

hypothetical deduction

my mind needs more RAM to handle these long-chain hypothetical deductions TvT

A deduction which we most likely will have to backtrack on.

If solving Skyscrapers ‘properly’, this means we are ‘imagining’ the deductions in our mind. If they lead to a contradiction, then we can rule out this hypothetical solution as invalid.

lane

Now that we’ve found where the $6$-skyscraper goes in this lane, solving the rest of the lane becomes much easier.

A general term for either a row or column of a puzzle, as well as the clues adjacent to it.

For instance, this is the lane $\text{5 | 1 2 3 4 5 | 1}$:

See also: half-lane

lane peak

An $N$-skyscraper in a NxN Skyscrapers puzzle.

look

If we look across this half-lane, we need to be able to see four skyscrapers.

“Look” is used abstractly to describe considering the cells in a lane.

Pretty difficult to describe technically, but I’m sure you already understand it! For an intuitive explanation, please see the Skyscraping primer.

obscured

In contrast to a visible skyscraper.

open

I can solve closed 7x7 Skyscrapers with time, but it takes ingenuity to finish an open 7x7 Skyscrapers.

In contrast to a closed Skyscrapers, an open Skyscrapers does not have clues along every lane, but only a select few. This leaves parts of the edges of the grid ‘open’.

With fewer but impactful clues, these Skyscrapers are much more ‘directed’ and challenging.

Also called “sparse” for historical reasons.

peak

We’ve found 3 peaks already, but the clue is $4$, so there must be 1 more peak before these 3 peaks.

A skyscraper that is guaranteed to be visible.

Akin to a “maximum” in mathematics.

pencilmarks

This cell could contain a $1$ or $2$, so we’ll write those candidates in as pencilmarks.

Small numbers placed in a cell to indicate what candidates could go in there:

puzzle

Refers to the entire Skyscrapers puzzle, including the grid and clues.

sequence

Since the clue is $4$ and we only have 3 unsolved cells, this half-lane must have a sequence.

An ascending sequence of skyscrapers, when looking across a lane from the head cell towards the lane peak.

This usually refers to a series of unsolved cells with the constraint that they must be ascending. For instance, $\text{| [12] [23] [34]}$ is a sequence that could be solved to $\text{| 1 2 3}$, or $\text{| 2 3 4}$, or any mix such as $\text{| 1 3 4}$.

$\text{| [12] [123] [123]}$, in contrast, would not be a sequence, since it’s not necessarily ascending – it could end up being solved to $\text{| 2 1 3}$.

skyscraper

The $5$-skyscraper must go in this cell.

A number inside a cell of a Skyscrapers puzzle. As opposed to the a number outside the grid, which is a clue.

For instance, a 4x4 Skyscrapers contains $4 \times 4 = 16$ skyscrapers.

$n$-skyscraper refers to the skyscraper of height $n$ (i.e. the number $n$). In an NxN Skyscrapers, the $N$-skyscraper refers to the tallest skyscraper of height $N$.

Skyscrapers

We love solving Skyscrapers!

A Skyscrapers puzzle. It’s capitalised since it refers to the name of the puzzle, not a skyscraper (number) in the puzzle.

To avoid ambiguity I may sometimes use the fully qualified form “Skyscrapers puzzle”.

solved cell

We’ve got loads of pencilmarks, but not many solved cells =(

A cell which has been definitively assigned a skyscraper.

In contrast to an unsolved cell.

sparse

Another name for an open Skyscrapers.

tail cell

Since the tail cell is a $6$ and the clue is $2$, the head cell must be a $5$.

The last cell in a lane.

In this lane, the tail cell is the one containing the $4$-skyscraper.

See also: head cell

unsolved cell

A cell which has not been definitively assigned a skyscraper, and may have multiple candidates indicated with pencilmarks.

In contrast to a solved cell.

visible

In contrast to an obscured skyscraper.