Usowan

Every number is two statements, and you do not get told which

Usowan prints numbers and then, in the same breath, says that one number in every region is wrong — without saying which. So a clue is never the sentence “my count is v”. It is “my count is v, unless I am my region's liar, in which case my count is anything but v”, and the second half of that is still information. Both halves can be priced exactly, because the whole answer space of a 6×6 board fits in memory: 1,646,096 shadings with no two shaded cells touching and the white cells in one sheet. Walk all of them, ask how many shaded cells sit beside a given cell, and the two readings of every printed value fall out.

The ranking is exactly inverted between the two readings, which is the whole trick of the genre. The sharpest number that can be true is 3 — it survives in only 3.16% of the answer space — and the same 3, read as the lie, rules out almost nothing at 96.84%. A 0 is the blunt one honestly, leaving 45.58%, and the sharp one as a lie, leaving 54.42%. Since the board never says which reading applies, every number you look at is simultaneously the strongest and the weakest thing in front of you, and the only way through is to work out which one in this region is lying before you can use any of them.

One value per cell is a lie before you look at anything

The last row of that table is a hard zero, and it is not a rounding. A clue that prints as many as it has neighbours — a 4 in the middle, a 3 on an edge, a 2 in a corner — would have to sit in the middle of a fully shaded cross. Its own cell is white, because numbered cells always are, and it would then have no white neighbour at all: the white sheet would be torn. So that clue cannot possibly be honest, which makes it its region's liar, which settles every other number in the region at a glance. It is the only clue on a Usowan board that tells you nothing whatsoever about its own four neighbours and everything about its neighbours' numbers.

That the white-sheet clause is what does it, rather than the no-touching clause, is measurable too. Count the same 6×6 space twice — once with the sheet clause and once without — and the impossible values are impossible in exactly one of the two.

The generator does not go looking for those clues and does not avoid them either — it picks each region's liar at random and gives it a random wrong value — so how often they turn up is a property of the genre, not of this repository. Across the 1,393 lies on the 108 shipped boards, 404 of them (29.0%) are of this self-announcing kind.

How much room there is, before a single number is read

An answer, stripped of every number, is a set of shaded cells that are pairwise non-adjacent and whose complement is connected. src/count.ts counts those exactly with a broken-profile sweep: one cell at a time, carrying a frontier that records, for each column, whether the cell is shaded and — if it is white — which white component it currently belongs to. Because the sweep is row-major on a planar grid those components never interleave, so the state count stays small. Drop the connectivity clause and the very same sweep counts the independent sets of the grid graph, which is somebody else's sequence and therefore a check rather than a claim.

The clause that the rules spend four words on — all the white cells are connected — is the expensive one, and it gets steadily more expensive as the board grows: 2.19× at 4×4, 5.55× at 8×8, 84.4× at 16×16. The sequence of answer counts — 1, 5, 39, 562, 20,297, 1,646,096, … — returns nothing from the OEIS in September 2026. The right-hand column does: it is A006506, the independent vertex sets of the n×n grid graph, and the sweep reproduces all twelve published terms with no special-casing at all.

A board is a stack of ordinary puzzles, one of which has an answer

Name one number per region as the liar and Usowan collapses into an ordinary shading puzzle: every other number is honest, that one is known false, and nothing is hidden any more. A board whose regions carry k₁, k₂, … numbers is therefore k₁·k₂·… ordinary puzzles superimposed. The median 6×6 in this bank is 216 of them (the widest is 864), and because the board has exactly one answer, exactly one of those readings is solvable and the rest are contradictions waiting to be found. Which clause finds them is the part worth measuring.

Over all 10,335 readings of the 36 boards small enough to take apart exhaustively, 99.6% die to plain counting and only 9 of them need one of the two structural clauses. The interesting number is the last one: 36 readings survive everything, one per board, which is the same statement as “the board has exactly one answer” arrived at from the other end.

Printing one more true number can give the board a second answer

In every other shading puzzle a number is a monotone constraint: print one more true one and the answer set can only shrink. Usowan is not like that, and the reason is the same sentence that makes the genre. A region's rule is exactly one of my numbers is wrong, so an extra number is an extra candidate liar. A shading that used to be rejected because the region had nothing wrong in it can become legal the moment there is a new number for it to be wrong about.

That is measurable on the boards this page ships. Take a board, pick any white cell that prints nothing, and print its honest count on it. The intended answer stays legal every time — it has to, the region still has its liar. Everything else is up for grabs.

Across all 2,391 true numbers that can be added to the 108 shipped boards, 55 of them hand the board a second answer and 6 more take it out of reach of the ladder without costing it uniqueness. On 20 of the 36 10×10 boards there is at least one cell where writing down something true breaks the puzzle. This is also why the generator has to rub numbers out to a fixed point rather than in a single pass: a number that is load-bearing while its neighbours are still printed can become removable once they are gone, and one pass leaves boards that are not minimal.

The shading hugs the border, and the blank board says so exactly

Run the sweep forwards and then backwards and every frontier state learns how many answers can still be finished from it. Forward weight times backward count is the number of answers passing through, so the probability that a given cell is shaded comes out exactly — over the entire space, not over a sample of it.

On an 8×8 a corner is shaded in 31.06% of all 118,930,018,898 answers and the middle of the board in 20.69% — the corner wins by 1.50×. Both clauses push the same way. A shaded cell in the middle has to keep four neighbours white where a corner keeps only two, and it is also the cell with the most ways to pinch the white sheet in two — a corner can only ever cut off the corner. The cheap real estate is the border, and the answer space knows it long before any number is printed.

The second grid is the 36 8×8 boards this page ships, and the honest reading of it is that 36 boards are not enough to read anything from. The largest per-cell gap against the exact probabilities is 17.2%, which is 2.2 times the one-sigma sampling noise for a bank this size (7.7% per cell) — and it is the worst of 64 cells, so it is not a number to read anything into. There may well be a selection effect hiding in there: the answers are uniform, but only the ones that turn out to be solvable get shipped, and nothing guarantees that filter is neutral. This bank cannot show it. The left-hand grid is not a sample at all — it is the whole space, and the border really is brighter than the middle.

The generator draws its answers from that same machine — walking the frontier forwards with each choice weighted by how many finished answers it leaves reachable, so no rejection and no retries — and that the draw is uniform is checked rather than assumed.

The ladder

• cell — the board with no number read at all. A numbered cell is white; the four neighbours of a shaded cell are white. That is the whole rung.
• count — the numbers, as bounds and as bookkeeping. For each region, work out which of its clues could still be the liar — a clue whose value is out of reach is disqualified, and so is one that has already come true. Any clue that is honest under every surviving reading gets saturated; if only one candidate liar is left, its count is forced to miss the number it prints.
• region — stop approximating the disjunction and solve it. Take the union of the region's clue neighbourhoods, enumerate every shading of it that keeps shaded cells apart and leaves exactly one number wrong, and keep what they all agree on.
• connect — the white cells are one sheet. An unknown cell cut off from every white cell can never be white, so it is shaded; an unknown cell whose removal would tear the sheet can never be shaded, so it is white. One Tarjan pass finds every such cell at once.
• probe — assume a cell, run the cheaper rungs, and drop the assumption if the board dies.

Every shipped board is finished by connect by construction — the generator kept rubbing numbers out only for as long as propagation could still close the board without guessing — so what the table shows is the shape of the climb. At 10×10 the board's own geometry settles 46.0% of the grid and finishes nothing; reading the numbers as bounds takes it to 79.5%; solving each region's disjunction exactly takes it to 96.6% and finishes 9 boards; the white sheet closes the other 27.

The same rungs, priced as pruning instead of as proof

Run the complete search, but only let it propagate up to a given rung between branch points, and count the branch points. The ceiling is 200,000; a board that hits it is reported as capped rather than quietly averaged in.

The gap between the first two rows is the whole genre. Without the numbers a 6×6 takes a median of 29,123 branch points; with them read only as bounds it takes 2. Neither rung knows which number is lying — the count rung only knows which ones could be — and that is already enough to take four orders of magnitude off the search. At 8×8 the bottom rung does not finish inside the ceiling on a single board.

The shipped boards

Roughly 40% of the numbers on a shipped board are lies, and none of them can be rubbed out. Deleting a lie leaves its region with no wrong number, which makes the intended answer illegal — so the lies are the load-bearing part of the board by construction, and only the honest numbers were ever candidates for removal. Those were rubbed out in a random order for as long as the ladder could still close the board, which is why removing any one of the survivors breaks it. The tests re-derive both halves of that rather than trusting them.

Four ways to misread the board

They split two and two. The generous pair only ever adds answers, so the intended answer stays legal and nothing looks wrong until the very end: letting a region get away with no lie at all leaves 8 of 36 8×8 boards unique, and letting white connect through a corner leaves 13. The other pair takes answers away instead. Reading every number as honest, and reading the board as carrying one lie in total rather than one per region, both make the intended answer illegal on all 36 boards at every size — a board with 12 regions is already carrying 12 lies. That is the better failure mode: those players do not wander off into a second answer, they get stuck, and getting stuck is something you notice.