Nondango

Two of the three circles can live in the same region, and then the rule says nothing

A forbidden set is three circles in three consecutive cells of a row, a column or a diagonal. But a region only ever paints one circle black, so if two of those three circles sit in the same region they can never both be black, and the set is satisfied before the puzzle starts. It is dead. On the boards shipped here that is 34.8% of all the geometry — an 8×8 draws 43.1 lines of three circles and only 28.3 of them can ever fire; a 12×12 draws 146.4 and keeps 95.3.

The screen is the first thing the solver does and it is not an approximation: on all 36 of the 8×8 boards, solving with every geometric triple and solving with only the live ones return the same answer set, 36 times out of 36.

One swap is all it takes

Take the answer and move one region's black circle onto a different circle of the same region. Every other region is untouched, so it is still one black per region — the only thing that can make it illegal is a line of three through the circle you just painted whose other two circles were already black. Call that line the circle's certificate.

So a board with one answer needs a certificate for every circle the answer leaves white. Checked the blunt way: all 2,113 such moves across the 72 shipped boards produce an illegal grid, and all 2,113 of them have a certificate to explain it.

Now count the certificates. A line with two black circles has exactly one white one, so it can certify that circle and no other. Therefore

circles − regions ≤ certifying lines ≤ live triples

and the left-hand side is the entire amount of choice on the board. This is the budget the puzzle has to work with, and it is not generous: the tightest shipped board has just 1 certifying line to spare (median 9), and 0 boards break the inequality.

No live triple, no puzzle at all

Join two regions when a live triple touches both. The rule never crosses from one connected group to another, so the answers of the groups are independent and the total is their product — which is why the complete solver is just a backtracker run once per group. A region in no live triple is a group of one, and it contributes a factor equal to its number of circles: free choice.

Measured rather than assumed: of 200 random boards that happened to have no live triple at all, 200 had exactly the product of their region sizes as their answer count. The 72 shipped boards split into 189 groups, 114 of which are single regions holding a single circle — the givens — with the rest of the board collapsing into one group of up to 48.

Both ways of misquoting "three in a row" delete the puzzle

The rule is usually written "three circles in a row, column or diagonal", and the two words people drop are opposite mistakes.

Read row as excluding the diagonals and the constraint graph loses most of its certificates. Every one of the 72 shipped boards stops being unique: 0 of 36 at 8×8, with a median of 1,268 answers, and all 36 of the 12×12 boards run past the 20,000-answer counting cap.

Read in a row as "anywhere in the same line" instead of "in consecutive cells", and the puzzle goes the other way: on a line of L circles the legal black sets fall from the tribonacci numbers, A000073, to at most two blacks per line — 1 + L + L(L−1)/2, the lazy caterer's sequence, A000124. Exponential becomes quadratic, and 72 of 72 shipped boards have no answer whatsoever.

For the record, the same count over a whole square grid where every cell carries a circle is A181218. The transfer matrix below is checked against brute force while brute force can still keep up, and stops at 8×8 because the 9×9 term is past 253 and a double would quietly round it.

The dial is how many white circles you dare add

The generator draws the regions, then the answer, then adds white circles one at a time — but only where a certificate already exists, because nothing else could ever be ruled out. Certification is necessary. The dial measures how often it is also sufficient: k certified circles thrown in at random, with no uniqueness check at all.

That is 8×8. A 12×12 has more room and the same shape of curve, from 100.0% at 0 circles down to 13.7% at 32. Note what never happens: the answer count goes up, never to zero. Adding a circle cannot invalidate the answer, only give it company — 0 of 3,276 dial boards had no answer.

The shipped boards keep every circle the solver can still eliminate, which lands at 15.4 white circles on an 8×8 and 43.3 on a 12×12: 15,849 ways to paint the 8×8, one of which survives, and 10^11.6 ways to paint the 12×12.

Boards do not happen by accident

Random regions, random circles, then ask for the answers:

2,248 of 3,600 have no answer at all, 27 have exactly one, and only 23 of those hold more circles than regions — the rest are boards where every region was handed a single circle and there was nothing to solve. Uniqueness is not rare because it is delicate; it is rare because it has to be built.

The ladder

Four rungs over the same per-region domains.

• region — one black per region and nothing else. Decides the regions that hold a single circle.
• triple — hyper-arc consistency on the live triples, to a fixpoint. This is the certificate rule read as propagation: once two circles of a live triple are pinned, the third is out.
• probe — singleton consistency: assume one circle, propagate, drop it if that alone contradicts.
• search — the complete solver, one component at a time.

The split tracks the amount of choice on the board almost exactly. Boards generated at 8×8 and graded by how many white circles they ended up with:

triple is unit propagation over certificates, so it needs somewhere to start: a certificate whose two black circles are already pinned. It stalls when the remaining circles all certify each other in a loop. That is what 52 of the 72 shipped boards do — and even there triple alone still pins 72% of the regions before it gets stuck, leaving a knot of a median 13 regions for probe. Nothing in the bank needs search.

What I expected and did not find

A certificate needs two black circles in adjacent cells, so the answers ought to look clustered — black circles leaning on each other everywhere without ever making three. They do not look like anything in particular. Counting the black circles with at least one black neighbour among the eight: 86% on the shipped 8×8 answers against 87.8% for a black circle picked at random in each of the same regions, and 91% against 91.4% at 12×12. At these region sizes the geometry hands out adjacent pairs for free and the generator never has to ask for them.

Two engines

The dead-triple screen and the component split are the load-bearing claims here, so both are checked against an engine that has neither: a plain backtracker that picks one circle per region by exhaustion and then walks the finished grid against every geometric triple, three cells at a time. On the boards small enough for it to finish the two return the same answer set, not just the same count, and the same holds under both misreadings. The component split is checked a third way, by counting each group by hand and multiplying.