The solver said no 6×6 was easy

Skyscrapers is the last puzzle type on Puzzle Lab's list. A Latin square — every row and column holds each height once — clued from outside: a number on an edge says how many towers you can see looking in from there, taller ones hiding shorter ones behind them. No givens, no boxes, just the edge numbers.

An animation of a 5×5 Skyscrapers board on Puzzle Lab being filled one hint at a time. Clue digits sit outside the dark grid on all four sides: a 3, a 1 and a 3 above, a 2 and a 4 below, a 2, a 1 and a 2 on the left, a 3 and a 1 on the right. Each frame places one more digit, highlighted in gold, and a Hint line under the numpad says why: clue 1 on column 3 from the top means only one tower is visible, so the tallest, 5, stands next to the clue; then 5 has only one place left in row 2; then only 4 fits at row 1, column 5; and so on through twelve placements.

That is the finished thing above, explaining itself one hint at a time — every placement named by the technique that found it. The grading those names feed is what this post is about.

I built it the way I built Kakuro a day earlier: a research pass, a plan with numbered slices, and the rule that the visual surface lands first on hand-baked puzzles while the engine goes in underneath. The exact solver, the human-style solver that grades difficulty by the hardest technique a puzzle needs, the generator — each its own small pull request, each reviewed before the next.

It went to plan right up to the slice whose whole job is to find out whether the plan works.

The yield spike

Before writing a generator I measure what one would produce. The design was Tatham's: fill a random Latin square, derive all 4N edge clues, then blank clues one at a time while the puzzle stays unique and the solver can still finish it at or below the tier you want. Remove toward "easy", accept when the grade is easy.

Random squares are rarely unique even with all their clues — a third of 5×5 squares, one in twelve at 6×6, none of 94,962 random fills at 7×7 — so there is a repair step first (swap a 2×2 sub-square's corners, keep it if the solution count does not rise). That part behaved. The tier-bounded removal did not:

target tier at 6×6	easy	medium	hard	expert	extreme
puzzles landing exactly there	0%	0%	85%	10%	73%

Forty attempts per tier, zero easy, zero medium. Every attempt aimed at easy landed on hard or extreme instead.

Removal only goes up

The thing I had not thought through: blanking a clue can only make a puzzle harder. So the grade of the fully clued square — every one of its 24 numbers showing — is a floor, and removal can never reach a tier below it. I measured the floor directly, 300 unique squares with all their clues:

tier of the all-clue 6×6	easy	medium	hard	expert	extreme	beyond the ladder
squares of 300	1	1	273	2	20	3

Ninety-one percent of random 6×6 squares graded hard before a single clue came off. That cannot be right. Published easy Skyscrapers are fully or nearly fully clued and beginners solve them. Whatever a beginner does with a fully clued line, my ladder was calling tier 3.

What a beginner actually does

The ladder's tier 3 held a technique I had called line filtering: take one row, enumerate every arrangement of heights that satisfies its two clues, and strike any height no arrangement allows. It is the catch-all — Tatham's "hard" solver does exactly this — so I had filed it as hard.

But "enumerate every arrangement" covers two very different amounts of work. I instrumented the solver to record, at each line-filter step, how many arrangements of that line were still possible when it fired:

A bar chart titled How much work was the line scan. For each count of arrangements still possible when the scan fired — 1, 2, 3, 4, 5, 6, 7 to 12, 13 to 24, and 25 or more — two bars show how many scans fell there: orange for 5×5, 857 scans over 300 squares, and purple for 6×6, 1,860 scans over 200 squares. The 5×5 bars are 310, 178, 219, 46, 46, 15, 41, 2 and 0. The 6×6 bars are 245, 167, 261, 160, 118, 227, 391, 164 and 127. Dashed lines split the chart into three shaded bands labelled tier 1, up to 3 arrangements; tier 2, up to 12; and tier 3, more than 12. A caption reads: a third of the 5×5 scans looked at one arrangement; the old ladder graded that the same as a 60-arrangement search, hard.

A third of the steps a 5×5 needed were scans of exactly one arrangement, and most of the rest of two or three — "a 1 facing a 2 on this four-cell row: the 4 goes first, the 3 last, and only two orders remain." That is the first move every guide teaches. It is not a hard technique. The ladder had one word for a one-arrangement glance and a sixty-arrangement search, and graded both as the search.

The fix was two numbers

The scan is now three techniques, graded by how many arrangements it had to look through: up to three is tier 1, up to twelve is tier 2, more is the old tier 3. Nothing about the puzzles changed. Their grades did:

6×6, target tier	easy	medium	hard	expert	extreme
before the re-tier	0%	0%	85%	10%	73%
after	28%	85%	58%	15%	53%

Two stacked horizontal bars titled Same 300 squares, graded twice, each spanning the 300 random 6×6 Latin squares graded with all 24 clues showing. Before, with a flat tier-3 line filter: a thin sliver of easy and medium, then hard filling 91 percent of the bar, then a small expert slice and extreme at 7 percent. After, with the scan graded by arrangements considered: easy 25 percent, medium 61 percent, a thin hard slice, a thin expert slice, and extreme 9 percent. A caption reads: before, 273 of 300 graded hard before a single clue came off; after, 1 in 4 easy, 3 in 5 medium; not one puzzle changed.

The all-clue floor at 6×6 went from 91% hard to 25% easy and 61% medium, and the hand-made 5×5 that had carried every UI slice — graded hard by the old ladder — turned out to be easy.

The cuts are provisional and the tier calibration slice will refit them against the scorer. But the lesson is older than the numbers. A difficulty classifier is a model of a person. When a technique is a catch-all, the tier has to come from the size of the work, not the name of the rule.

Two smaller things the spike caught

The 7×7 repair climb was failing half the time, and the plain reading was "7×7 is marginal". It was not. A random 7×7 starts with sixty or more solutions, and sixty was also the cap on the solution count the climb minimises, so every swap read "60 → 60" and the climb wandered. Restarting from a fresh square after forty fruitless swaps, with the cap lowered to twenty, brought every attempt home in a median of 178 milliseconds. When a hill-climb stalls, check whether its objective is flat before deciding the hill is steep.

And the daily's standard board is not 9×9 any more. Every type until now had a 9×9 standard. The only Skyscrapers size that offers all five tiers — the 5×5 tops out at hard, the 7×7 has no easy — is the 6×6, so that is its standard slot. "A board is a mini if it is smaller than 9×9" was never written down as a rule, but four places had quietly assumed it, and the type-checker could not find any of them. Grep did.

Where it landed

Skyscrapers ships at 5×5, 6×6 and 7×7, every puzzle generated fresh at exactly the tier you pick, graded by the same solver that powers the Hint button. The whole offered table generates in under a second per cell — 6×6 easy through hard in 30–60 milliseconds — and a soundness fuzz over 1,500 generated puzzles found no placement the solver got wrong. It joins the daily as the fifth type, so a day is now eight boards and every difficulty is played every day.

Twelve pull requests in two days, with a review on each. The one that mattered most was the one with no code in it.

The cuts at three and twelve are mine, from the histogram. If you set Skyscrapers for people and would draw the easy/medium line somewhere else, I would like to know where, and why.