Heyawake
Play Heyawake — and see why the rule everyone forgets is load-bearing: drop it and 362 uniquely solvable layouts collapse to just 20.
Click a cell to cycle blank → dot → shaded. Dots are your own notes. A number says exactly how many cells of that room are shaded. No two shaded cells may touch side by side, all unshaded cells must stay joined up, and no straight line of unshaded cells may pass through more than two rooms. Everything runs in your browser; nothing is uploaded.
The rule everyone forgets is the one holding the puzzle up
Heyawake's third rule — no straight run of unshaded cells through more than two rooms — reads like decoration. It is the fiddliest thing to check by eye, it never says anything about a single cell, and plenty of solvers only reach for it once they are stuck. So: would the puzzle still work without it?
It would not. Below, every room is given its true count — the most generous setting there is, nothing hidden — and each layout is then solved twice, once with the rule and once without:
| Board | Rooms | Layouts | One answer, with the rule | One answer, without it |
|---|---|---|---|---|
| 6 × 6 | 6 | 207 | 62 (30%) | 8 (4%) |
| 6 × 6 | 8 | 200 | 111 (56%) | 6 (3%) |
| 6 × 6 · recomputed | 10 | 183 | 115 (63%) | 3 (2%) |
| 7 × 7 | 8 | 72 | 14 (19%) | 0 (0%) |
| 7 × 7 | 10 | 159 | 60 (38%) | 3 (2%) |
Across all five rows that is 362 uniquely solvable layouts with the rule against 20 without — a factor of 18. And it is not that the answer count creeps up to two or three: without the rule the median layout runs past the counting cap of 40. Whatever that rule is, it is not decoration.
The first version of that test was rigged, and looked far better
The obvious way to run this experiment is to make a proper puzzle — clues pared down to the minimum, the way a real one is printed — then take the rule away and see what survives. Done that way, 0 of 154 puzzles stayed unique. A perfect score, and worth nothing.
Paring the clues down with rule C active and then solving without it left 0 of 154 puzzles unique. That is not evidence: the clue set had been trimmed to lean on C, so removing C afterwards was bound to break it. The fair test clues every room in both arms.
The honest version is the table above, and it comes out weaker — a factor of 18 rather than everything-to-nothing. That is generally how it goes: the rigged version of a comparison is the one that gives the cleaner number.
What the rule is worth in clues
Measured the other way round: dig each puzzle down separately in each arm, using the same removal order so the two are matched, and see how many clues each needs.
| Board | Rooms | Puzzles paired | Clues needed with the rule | Without it | Difference |
|---|---|---|---|---|---|
| 6 × 6 | 6 | 8 | 3.88 | 5.00 | +1.12 |
| 6 × 6 | 8 | 6 | 3.50 | 5.67 | +2.17 |
| 6 × 6 | 10 | 3 | 1.67 | 6.00 | +4.33 |
Only puzzles that stay unique in both arms can be paired, and those are exactly the ones where rule C is doing the least work. The saving shown is therefore an understatement, and rests on 3 to 8 puzzles per row.
Why the boards stop at 7 × 7
Not because larger Heyawake is unreasonable — printed puzzles are routinely bigger — but because the solver here checks every shading, and that gets expensive fast:
| Board | Rooms | Time for one puzzle | Clues kept |
|---|---|---|---|
| 6 × 6 | 10 | 10 ms | 3.1 of 10 |
| 7 × 7 | 12 | 240 ms | 3.4 of 12 |
| 8 × 8 | 12 | 54 s | 5.0 of 12 |
An 8 × 8 takes about 54 seconds a puzzle, which is not something to put behind a button. Worth noticing in that last column: only three or four rooms out of ten or twelve keep their number. Most rooms end up blank, and the puzzle still has exactly one answer — which is another way of saying how much of the work the three rules are doing on their own.
How to use
- Pick a board size and press New puzzle.
- Click a cell to cycle blank, dot, shaded. Dots are your own notes.
- A number in a room says exactly how many of its cells are shaded.
- No two shaded cells may touch side by side, and all unshaded cells must stay joined up.
- No straight line of unshaded cells may pass through more than two rooms.
Frequently asked questions
What are the rules of Heyawake?
The grid is divided into rectangular rooms. Shade some cells so that no two shaded cells touch orthogonally, every unshaded cell is connected to every other, and no straight horizontal or vertical run of unshaded cells passes through more than two rooms. A number in a room gives exactly how many of its cells are shaded; rooms without a number are unconstrained.
Does the two-room rule actually matter?
Enormously. Clue every single room with its true count and solve each layout twice: with the rule, 362 of 821 layouts had exactly one answer; without it, 20 did. Dropping the rule does not add an answer or two either — the median layout runs past a counting cap of 40 solutions.
Why do so few rooms have numbers?
Because the three rules do most of the work. Digging clues out for as long as the puzzle keeps a single answer typically leaves only three or four numbers out of ten or twelve rooms. The blank rooms are not missing information, they are rooms the rules already pin down.
What does the two-room rule mean exactly?
Take any straight line of consecutive unshaded cells, horizontal or vertical. Count how many different rooms it passes through. If that is three or more, the shading is illegal. It is the fiddliest rule to check by eye, which is why solvers tend to reach for it last.
Why do the boards here stop at 7x7?
Speed, not principle. The generator checks every shading, so a 6x6 takes about ten milliseconds and a 7x7 about a quarter of a second, but an 8x8 takes roughly fifty seconds — too slow to sit behind a button. Printed Heyawake is routinely larger.
Is Heyawake the same as Hitori or Nurikabe?
They are all shading puzzles with a connectivity rule, but the constraints differ. Hitori shades duplicate numbers in rows and columns; Nurikabe builds islands of given sizes in a connected sea. Heyawake is the one with rectangular rooms and the run-length rule across room boundaries.
Does this send anything anywhere?
No. Puzzles are generated and solved entirely in your browser, and nothing is uploaded.
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