Kurodoko Puzzle & Packing Limits
Shade cells so every number sees exactly that many — and see why the checkerboard, the best possible packing of black cells, is illegal.
Shade cells black. A number counts every white cell it can see in the four directions, itself included. Black cells may never touch side by side, and the white cells must all stay joined up. Click a blank cell to shade it.
The checkerboard is the answer to the wrong question
Two rules govern the black cells: no two may touch side by side, and every white cell must stay connected to every other. The first rule alone has a famous answer — a checkerboard, which fits ⌈n²/2⌉ black cells and cannot be beaten. And it is illegal: it leaves the white cells touching only at their corners, so the white region falls into isolated squares and the second rule kills it outright.
| Board | Squares | Checkerboard | Actually legal | Lost to connectivity |
|---|---|---|---|---|
| 3 × 3 | 9 | 5 — illegal | 4 | −1 |
| 4 × 4 | 16 | 8 — illegal | 5 | −3 |
| 5 × 5 | 25 | 13 — illegal | 9 | −4 |
| 6 × 6 | 36 | 18 — illegal | 12 | −6 |
| 7 × 7 | 49 | 25 — illegal | 17 | −8 |
By 7×7 the connectivity rule has taken eight cells off the naive answer — a third of them. The constraint that decides Kurodoko is not the one people reach for.
No formula is offered for that fourth column. The values run 1, 4, 5, 9, 12, 17 for boards from 2 to 7, and the tidiest candidate rule — ⌈n/2⌉² — fits at 2, 3 and 5 and is wrong at 4, 6 and 7. They are reported as computed, by exhaustive search, and nothing is extrapolated past 7×7.
And a clue of 2n−1 settles two whole lines at once
A clue counts every white cell it can see in the four directions, itself included. The most it can be is a clear row plus a clear column, sharing one cell: 2n−1, which is 9 on a 5×5 and 13 on a 7×7. A clue that large forces its entire row and column white in one stroke, and it is the only clue that settles anything by itself.
The smallest is 1, which needs black on all four sides. That is perfectly legal on the touching rule — the four blacks meet only at corners — so a 1 is a real thing to look out for rather than a misprint. It does, of course, cut that cell off from everything else, which is exactly why the connectivity rule then forbids it.
How to use
- Shade cells black so each number sees exactly that many white cells.
- Count in all four directions, including the numbered cell itself.
- Black cells may never touch side by side.
- Every white cell must stay joined to every other.
Frequently asked questions
What are the rules of Kurodoko?
Shade some cells black so that every numbered cell can see exactly that many white cells, counting in the four directions and including itself. Black cells may never touch side by side, and all the white cells must remain connected to each other. Numbered cells are always white.
How many cells can be black in Kurodoko?
Far fewer than the packing rule alone suggests. A checkerboard fits half the board and is the best possible answer to "no two touching", but it leaves the white cells connected only at their corners, so it is illegal. Exhaustive search gives real maxima of 4, 5, 9, 12 and 17 for boards from three by three to seven by seven.
Why is the checkerboard not allowed?
Because it disconnects the white cells completely. Each white square touches its neighbours only diagonally, and Kurodoko requires them to be joined side by side. The connectivity rule, not the touching rule, is what really limits how much of the board can be shaded.
Is there a formula for the maximum?
None is offered here, deliberately. The values are 1, 4, 5, 9, 12 and 17 for boards from two to seven, and the tidiest candidate rule fits three of those sizes and fails the other three. They are reported as computed by exhaustive search, with nothing extrapolated past seven by seven.
What is the largest possible clue?
Two times the board width minus one — a completely clear row plus a completely clear column, sharing one cell. That is nine on a five by five and thirteen on a seven by seven, and it is the only clue that settles anything on its own, because it forces its whole row and column white.
Can a clue be 1?
On the touching rule alone, yes: four black cells around a single white one meet only at their corners, so nothing is illegal about it. The connectivity rule then forbids it, because that white cell would be cut off from every other, which is a nice example of the two rules pulling in different directions.
Is Kurodoko the same as Kuromasu?
Yes, they are two names for the same puzzle. You may also see it called Where is Black Cells, which is a more literal translation of the original title.
Does every puzzle here have one answer?
Yes. Each generated puzzle is checked by counting its solutions and discarded unless there is exactly one, rather than trusting the shading it was built from.
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