ascendant

An automated solver for Skyscrapers^↗ puzzles, relying only on pure logic.

For a rundown of the algorithm, jump to § Algorithm. For an explanation of how Skyscrapers puzzles work, I have a quickfire explanation on Skyscraping^↗ you can check out.

Example

show

solving puzzle from https://www.brainbashers.com/showskyscrapers.asp?date=1217&size=5&diff=2

pass #0:
   |    4    |    3    |    2    |    2    |    1    |
---------------------------------------------------------
 4 | [12345] | [12345] | [12345] | [12345] | [12345] | 1
 3 | [12345] | [12345] | [12345] | [12345] | [12345] | 2
 2 | [12345] | [12345] | [12345] | [12345] | [12345] | 3
 1 | [12345] | [12345] | [12345] | [12345] | [12345] | 2
 3 | [12345] | [12345] | [12345] | [12345] | [12345] | 2
---------------------------------------------------------
   |    2    |    2    |    2    |    1    |    3    |

pass #1:
   |    4    |    3    |    2    |    2    |    1    |
---------------------------------------------------------
 4 | [   12] | [  123] | [ 1234] |    4    |    5    | 1
 3 | [   23] |    4    |    5    | [   12] | [   23] | 2
 2 |    4    |    5    | [ 1234] | [  123] | [  123] | 3
 1 |    5    | [  123] | [  123] | [  123] |    4    | 2
 3 | [  123] | [ 1234] | [ 1234] |    5    | [  123] | 2
---------------------------------------------------------
   |    2    |    2    |    2    |    1    |    3    |

pass #2:
   |    4    |    3    |    2    |    2    |    1    |
---------------------------------------------------------
 4 |    1    |    3    | [  123] |    4    |    5    | 1
 3 | [   23] |    4    |    5    |    1    | [   23] | 2
 2 |    4    |    5    | [  123] | [  123] | [  123] | 3
 1 |    5    | [   12] | [  123] | [  123] |    4    | 2
 3 | [   23] |    2    |    4    |    5    |    1    | 2
---------------------------------------------------------
   |    2    |    2    |    2    |    1    |    3    |

pass #3:
   |    4    |    3    |    2    |    2    |    1    |
---------------------------------------------------------
 4 |    1    |    3    |    2    |    4    |    5    | 1
 3 |    2    |    4    |    5    |    1    |    3    | 2
 2 |    4    |    5    |    1    |    3    |    2    | 3
 1 |    5    |    1    |    3    |    2    |    4    | 2
 3 |    3    |    2    |    4    |    5    |    1    | 2
---------------------------------------------------------
   |    2    |    2    |    2    |    1    |    3    |

Run

Clone the repository:

> git clone https://github.com/Sup2point0/ascendant

[!Important] You will need Rust nightly^↗ since the project uses generic_const_exprs.

See the algorithm in action:

ascendant> cargo run -- solve-one 7 --diff=2 --random-date

See what else you can do:

ascendant> cargo run -- --help

Features

Fetch puzzles:

# fetch 6x6 puzzles of all difficulties
ascendant> cargo run -- fetch --sizes 6 --diffs 1 2 3

Solve puzzles in bulk and view stats:

# solve all 4x4, 5x5, 6x6 puzzles of difficulty 3
ascendant> cargo run -- solve-all --sizes 4 5 6 --diffs 3

View the steps in solving one puzzle:

# solve the 7x7 full hard (difficulty 2) puzzle from April 1, showing all solution substeps
ascendant> cargo run -- solve-one 7 --diff=2 --date=0401 --debug

Progress

Note

Test puzzles sourced from brainbashers.com^↗, a wonderful puzzles site!

Full puzzles have clues along every lane in both directions. Sparse puzzles have much fewer clues, and are significantly more difficult. Bear in mind most human-oriented puzzles¹ are sparse 5x5 or 6x6 ;)

size	difficulty	solved	time	as of
4x4	full	365/365	~0.5 s	2026 January
	sparse	256/365	^	2026 January
5x5	full easy	365/365	~1.5 s	2026 January
	full hard	365/365	^	2026 January
	sparse	93/365	^	2026 January
6x6	full easy	359/365	~4.5 s	2026 January
	full hard	355/365	^	2026 January
	sparse	23/365	^	2026 January
7x7	full easy	331/365	~3.5 s	2026 January
	full hard	335/365	^	2026 January
8x8	full easy	321/365	~7.5 s	2026 January
	full hard	310/365	^	2026 January

Note

7x7 and 8x8 are taking less time currently since I haven't tried the solver on Sparse puzzles for those sizes yet!

Notes

show

• The algorithm is not strong at all on sparse puzzles, although it is fairly impressive how far it can get on sparse 5x5 puzzles.
  • Only 21/365 on the sparse 6x6 puzzles... 🤣
  • 4x4 sparse puzzles are quite specially intractable, simply down to how barren information is – some puzzles only have, like, 3 clues!
• Despite the increase in information with larger puzzle sizes, the increase in complexity is much more significant.
  • That being said, the algorithm often reaches an almost-solved state.
  • I’m currently figuring out how to improve the algorithm to handle these final near-solved states. Many of them need only a single push before the rules of Sudoku can clear the rest of the puzzle!
• I’m still torn over whether to implement a weak version of lane-isolated backtracking, because humans can do this mentally, so it's arguably not beyond the reach of “logic”. But it’s so ambiguous...

Algorithm

I affectionally call it peak descent!

Outline

The solving algorithm is iterative – it’ll keep passing over the grid, trying to make logical deductions, until it can no longer find any.

These are the steps in each pass-through:²

• Ascent: Use the clues to establish a foundation of candidates for each cell.
  • e.g. A lane $\text{4 | \_ \_ \_ \_ \_ \_ |}$ in a 6x6 puzzle could start with any of $[123]$, but not $[456]$.
• Peak Descent: Enforce ascending sequences by descending peaks.
  • If a peak has been found in a lane, step down from the peak towards the clue, calculating how many skyscrapers are currently guaranteed to be visible.
  • Subtract this from the clue to find how many more skyscrapers should be visible in front of the first peak.
  • Use this to restrict the candidates of the sequence.
    • e.g. In a lane $\text{4 | \_ \_ 4 \_ 6 5 |}$ this deduces $\text{4 | [12] [23] 4 \_ 6 5 |}$.
    • e.g. In a lane $\text{3 | \_ \_ \_ 5 \_ 6 |}$ this deduces $\text{3 | [4] [123] [123] 5 \_ 6 |}$.
• Sudoku: Eliminate invalid candidates by the rules of Sudoku.
  • e.g. A lane $\text{| 3 [36] [123] \_ \_ \_ |}$ can be eliminated to $\text{| 3 [6] [12] \_ \_ \_ |}$.
• Pinpoint: Mark cells which are the only place in their lane for a digit to go as solved.
  • e.g. A lane $\text{| [123] [23] [24] [34] 5 6 |}$ can be solved to $\text{| 1 [23] [24] [34] 5 6 |}$.
• Isolate: Eliminate candidates outside of ‘isolated’ groups (couples, triplets, etc.)
  • Two $[12]$ cells between them are guaranteed to use both $1$ and $2$.
    • We might not know which way round, but we can still deduce that $1$ and $2$ cannot go in any other cells in the lane.
  • This can be thought of as a ‘pseudo-pinpoint’ followed by sudoku elimination.
  • e.g. A lane $\text{| [12] [12] [123] [26] \_ \_ |}$ can be eliminated to $\text{| [12] [12] [3] [6] \_ \_ |}$.
• Solve: Mark cells with only 1 candidate left as solved.

Terminology

• Skyscraper: One number in a cell. For instance, the “5-skyscraper”.
  • Candidates: “Pencil marks” to indicate what skyscrapers could go in a cell.
• Lane: A straight line of cells in the $x$ or $y$ direction.
  • Row: A horizontal lane.
  • Col: A vertical lane.
  • Half-Lane: One side of a lane including cells up to the lane peak.
• Peak: A skyscraper guaranteed to be visible.³ (akin to a ‘maximum’ in mathematics)
  • Lane Peak: An $N$-skyscraper in an $N$×$N$ puzzle.
• Sequence: An ascending sequence of skyscrapers, looking from a clue across the lane towards the peak. Ideally strictly ascending, but not always so.

Development

Future strategies to implement include:

• Twin/triplet/... elimination: 2 $[xy]$ cells eliminate $x$ and $y$ as candidates from all other cells in the lane.
  • (more advanced) Closed set elimination: Any set of cells which between them consume $[xyz...]$ eliminate $[xyz]$ as candidates from all other cells in the lane.
• Solve cells with 2 candidates where one would be visible and the other obscured.

Philosophy

• The goal is to get as far as possible with purely logical deductions, i.e. no ‘guesswork’ or ‘backtracking’. Even though guesswork isn’t well-defined^↗...
• The strategies I plan on implementing are mostly those covered in Skyscraping^↗.
• If we stick firmly to this, then we may be able to also generate new skyscrapers puzzles!

Footnotes

1. After many years solving Skyscrapers (on-off, granted), I can speedrun some sparse 5x5, will need to spend maybe a good hour on sparse 6x6, and have only ever solved 1 sparse 7x7, which took many sittings over many days. Ofc, you may find easier large puzzles with specially designed tricks on sites other than Brainbashers, but this is the only site I’ve really used, so it’s all I have to go off =) ↩

2. The algorithm steps are executed in this order to optimise deduction rate, but you could theoretically use any order. ↩

3. Named this way because, if you were to look at the skyline of skyscrapers, you would see them taller than other buildings! ↩