Nawabari

The number tells you your seat, not the size of the room

Every cut-the-grid genre I had built before this one grades a region by a number that says something about the region — its area, its cell count, what its cells add up to. A Nawabari number says nothing about its region at all. It is a property of the cell it is printed in: how many of that cell's own four sides are walls. Which is to say it tells you which seat of the room the number is sitting in, and almost nothing else.

Counted over a 6×6 board — every cell, and every rectangle that cell could belong to — here is the whole alphabet. The column to watch is the fourth one: 4 is the only letter in the language that pins down how big its room is, and it does it by being the whole room.

Five letters, six seats — 2 is the one letter with two of them, and it cannot tell you which: the corner of a fat room and the middle of a 1-wide strip look identical on paper. That is the whole ambiguity budget of the genre, and the numbers above are the reason a board needs so many of them.

On the boards this page ships, a single number leaves a median of 6 rectangles open at 6×6 and 8 at 8×8 — spanning a median of 4 and 5 different room sizes, and up to 20. Nothing here is decided by one number. Everything is decided by how the numbers fence each other in.

The clue count is not a dial

This is the part that makes the genre strange to build. In every other puzzle I have shipped in this series the generator has a budget to turn: more numbers is easier, fewer is harder, and what is the smallest clue set that still works is a question with an answer. Nawabari does not have that question. One number per room, so the number of numbers is the number of rooms in the answer. The 72 boards here carry 633 numbers because their answers have 633 rooms, and there was never a choice about it.

What is left is where in each room its number goes, and that turns out to be the entire design space. Take a shipped answer, forget the printed board, and put one number in a random cell of every room: the value follows from the seat, so the board is always legal and always has that answer. Whether it has any other answer is the question.

So 19.5% of the ways to print a 6×6 answer produce a puzzle and 3.3% of the ways to print an 8×8 one do — 4,691 puzzles out of 36,000 placements drawn — and the rest produce a board with the right answer and at least one other. The spread across answers is the interesting part: at 6×6 the friendliest answer in the bank is unique in 84.3% of its placements and the most hostile in 2.3%, median 15.0%. At 8×8 the median answer is unique in 2.0% of its placements. The generator's job in this genre is not choosing what to say, it is choosing where to stand.

Same number, different cell

The sharp version of that experiment moves one number and leaves everything else alone. Take a shipped board, pick one of its numbers, and slide it to another cell of its own room, rewriting the value to match the new seat. The answer is untouched and still legal. Across the bank there are 2,967 such moves, and 1,485 of them — 50.1% — leave a board that is still unique. The other half break it.

And 1,360 of those moves do not change the printed number at all: the new seat has the same wall count as the old one, so the board looks like it is carrying exactly the same information in a slightly different place. Of those, 679 stay unique — 49.9%. Nothing about what the board says has changed, and it is a different puzzle. That is the separation this genre makes easy to see and most genres make impossible: here the position of a clue is not a delivery mechanism for its value, it is half of the value.

The seat the number lands on matters less than you would expect — every row is within a few points of half — with one exception at each end. A number that lands strictly inside a room (a 0) leaves the board unique 62.3% of the time, which is the best row in the table even though 0 is the weakest letter in the alphabet. A number that lands on a corner does worst, at 45.4%. Informative letters in obvious places are not what makes a board tight.

Erasing a number does not make a harder puzzle. It makes a non-board.

Every other genre in this series has the same local-minimum test: take any one clue away and check that a second answer appears, so no clue is decoration. The test cannot be run here, and the reason is worth stating precisely. Erase a number and the answer it belonged to is illegal, because one of its rooms now holds no number at all. There are 633 numbers in the bank, and erasing any one of them leaves the intended answer legal exactly 0 times.

So erasing a number destroys the board outright 59.4% of the time — and the other 40.6% of the time it leaves a board that still has answers, 144 of which are perfectly good puzzles with a different answer. In this genre a clue is not evidence about the answer. It is part of the specification of what an answer is allowed to look like.

The dots without the numbers

The complement of that experiment: keep every number's position and erase its value, so the board says "there is one number in each room, here is where they are". This is a real puzzle genre — it is Shikaku with the areas rubbed out — and the transfer matrix counts it exactly, with no cap, however large the answer gets.

0 of 72 boards survive with the values erased, and the median board goes from one answer to 564 at 6×6 and 369,090 at 8×8. Knowing where every number is, which is knowing one cell of every room, is not remotely knowing the answer.

Where a rule would put the numbers

If the seat were a formality, the generator could skip the search and print every number in, say, the top-left cell of its room. On the shipped answers that rule alone gets 39 of 72 boards unique.

The gap between the top-left rule and the bottom-right rule at 8×8 is not a fact about Nawabari — it is a fact about this generator, which grows rooms from the top-left corner and leaves the thin leftovers at the bottom-right, so the two ends of a board are not statistically alike. Drawing answers uniformly at random from all 535,236,230,270 dissections of a 6×6 instead, the asymmetry goes away: 50.2% of boards unique for top-left against 47.8% for bottom-right, over 600 answers each.

That uniform sample is worth one more line, because it says something about what a typical answer looks like. A dissection of a 6×6 drawn uniformly has 19.9 rooms, 9.8 of them single cells: the overwhelming majority of rectangular dissections are mostly 1×1s, which as boards would be mostly 4s and no fun at all. The answers here average 6.6 rooms at 6×6 and 10.9 at 8×8, because the generator weights each candidate room by its area when it draws.

Eight of the sixteen ways walls can meet

Everything in the rule except "one number per room" is a condition on a single lattice point. Four wall-stubs meet at an interior point; of the sixteen ways they can be walls, 8 are legal. A lone stub is a wall that borders nothing. Two perpendicular stubs make a reflex corner, so the room wrapped around it is not a rectangle. Nothing, a straight crossing, any of the four Ts, and the plus are all fine.

That is the entire geometry of the genre, and npm test checks the claim the hard way: over every one of the 2^12 wall patterns of a 3×3, every 2^13 of a 2×5 and every 2^17 of a 3×4, "legal at every lattice point" and "every room is a rectangle and every wall borders something" agree on every single pattern, and the surviving count is the published dissection count.

Across 2,664 interior lattice points in the bank the two illegal patterns appear 0 times, as they must, and the plus — four walls meeting, four room corners touching — appears only 11 times. Being local is what makes the genre countable: it is a vertex model, so a transfer matrix walks it column by column.

The ladder

Two of the rungs live on the edges and two live on the rectangles, and the handover between them is where all the strength is.

• count — a numbered cell has exactly as many walls around it as it says, and the outer frame is a wall. This is arithmetic on four bits.
• vertex — the eight legal patterns at a lattice point, propagated. This is the rung that turns a stub into a wall that goes somewhere.
• room — an answer is one rectangle per number, chosen so they tile the board. Filter each number's candidate rectangles against the walls known so far, read back whatever every survivor agrees on, and hand any cell that only one number can still reach to that number.
• probe — assume a rectangle, run the three rungs below, drop it if that alone contradicts.
• search — fewest surviving rectangles first, over the same candidates.

The two cheap rungs together settle 14.5% of the walls at 6×6 and finish nothing. The rectangle rung settles 98.3% and 89.4% and finishes 35 and 28 boards outright; probe finishes all 72. No board in the bank needs the search at all.

Which is easy to say and worth pricing. Run the complete search but only let it use the rungs up to a given level between branches, and count the branch points:

With only the two edge rungs the median 8×8 board costs 98,164 branch points. Add the rectangle rung and every board in the bank costs 0. The whole solver is that one change of view: stop asking which edges are walls and ask which rectangle each number belongs to.

Four ways to misread the rule, and they do not fail alike

The rule is four clauses — rectangles, exactly one number per room, the number counts its own cell's walls, the frame counts as a wall — and each can be got wrong on its own.

The two tightenings are loud: forget that the frame is a wall, or read the number as an area the way every Shikaku-shaped genre would, and the intended answer becomes illegal on 72 of 72 boards and the board goes completely blank. You cannot ship that mistake, because nothing works.

The loosenings are the dangerous ones, and the two halves of "exactly one number per room" are not worth the same. Dropping at most one costs uniqueness on 27 of 72 boards, which is real but survivable; the median board still has one answer. Dropping at least one — letting a room carry no number, which is exactly what the neighbouring genres allow — takes the median 8×8 board from one answer to 21,823,026,293 and the worst to 336,544,355,664,414. One half of one clause is holding the entire genre up.

A board drawn at random is almost always dead

The clue placement being pinned to the answer has a second consequence: a Nawabari board is an unusually delicate object. Scatter 7 numbers over a 6×6 at random and count the answers exactly:

Drawing the values from the distribution the shipped boards actually use — which is heavy on 3 and 2 — helps, and it does not help much: 99.0% of 8×8 boards drawn that way have no answer at all. Uniqueness has to be built from a known answer outward; there is nothing to find by scattering numbers.

Counting, against numbers somebody else published

With every number erased, a Nawabari board is a bare rectangular dissection, and those are counted in the literature. The transfer matrix has to reproduce A333476 as a triangle — 28 entries, all of them right — and then keep going down the diagonal, A182275, past anything a search could enumerate.

The rows of the same triangle are published separately — A034999 for 2×n, A208215 for 3×n, A220297 for 4×n, A220298 for 5×n — and 1×n is 2^(n-1), the compositions of n. Two searches that have never heard of the transfer matrix reproduce the corner of it they can reach: bruteByRooms, which lays rectangles down from the first uncovered cell, and bruteByEdges, which decides one internal edge at a time and only asks what the board looks like at the very end.

And one count nobody has published: which answers can be printed at all

A dissection is an answer. Whether it is a puzzle depends on whether some way of writing one number into each of its rooms leaves it alone, and that is a question about the dissection, not about any board. Walk every dissection of a small grid and every placement it admits:

The middle column, down the 1×n strips, is 7 terms of F(2n) — A001906, the even-indexed Fibonacci numbers — because the legal boards on a strip are the compositions of n weighted by the product of the parts. The column beside it, the ones that are unique, is 1, 3, 8, 21, 53, 133, 331, and searching the OEIS for that in September 2026 returns nothing; nor does the 2×n row, 17, 122, 879, 6169, 42505, nor 3×3 and 3×4 at 2,224 and 43,436.

The last column is the one I expected to be interesting and expected to be non-zero much earlier. Every dissection of every grid up to 3×4 can be printed as a puzzle: there is always somewhere to stand. Checking printability only needs one unique placement, so it stops early, and that makes larger grids reachable:

So the first unprintable answers show up at 4×4 — 12 of 70,878 dissections, one in 5,907 — while a 3×6, with 314,662 dissections and four and a half times as many chances to go wrong, has none at all. Size is not what does it. Shape is: every one of the 12 dead answers at 4×4 has exactly 6 rooms, every one of them is made entirely of 1-wide strips, and every one of them contains two parallel dominoes filling a 2×2 square. At 4×5 that stays nearly true — 247 of 267 are all strips and 260 contain the domino pair.

The domino pair is worth a second look, because it is the smallest ambiguity the genre has. Two dominoes side by side fill a 2×2 square, and in a domino every cell is an end, so all four cells print 3 whichever way the square is split. The numbers cannot tell the two halves apart; only which cells carry them can, since one number per room means one per row in one split and one per column in the other. On its own that is resolvable, which is why a 3×6 never dies. It is when a board is nothing but strips, and every escape from one ambiguity walks into the next, that an answer runs out of places to stand.