Sudoku Made and maintained by Brian Hamilton and Karan Hamilton.
Brian builds the playable puzzle tools, while Karan helps test clarity, difficulty feel, and the player experience.
Playable puzzles and controls are tested in the browser for rule accuracy, usability, and a clear solving experience. See how we make and test puzzles, our editorial standards, or report a problem.
What is Kropki Sudoku?
Kropki Sudoku, also called Dots Sudoku, is ordinary sudoku with dots on the boundaries between side-by-side squares. Fill it so that:
- every row, column and 3×3 box holds 1 to 9 exactly once;
- across a white dot the two digits are one apart, like 4 and 5;
- across a black dot one digit is double the other, like 3 and 6;
- and across a boundary with nothing drawn, neither is true.
1 and 2 are both one apart and double-and-half, so that pair could carry either dot; on this page it is always black. There are 144 boundaries on the grid and a board here carries about 44 dots, so around a hundred boundaries are blank — and every one of them is a clue.
Most variants only tell you what is drawn. Kropki, as it is usually set and as it is played here, also tells you what is not: every pair of neighbours that is one apart or double-and-half gets its dot, so a boundary with nothing on it says the two digits are neither. A 4 with nothing between it and its neighbour rules out 2, 3, 5 and 8 next door. If you know Killer Sudoku, you already play with clues that are not digits; kropki goes one step further and makes the absence of a clue count too. Some setters publish puzzles where only some dots are shown and say so in the rules; when a puzzle does not say, assume every dot is there, as it is on every board this page deals.
What does a boundary tell you before you place anything?
Read the shapes first. 5, 7 and 9 can never touch a black dot, because none of them has a double or a half between 1 and 9, and 1 can never touch a white dot, because the only pair it could make there is 1 and 2, which is drawn black. A square with black dots on both sides of one line — left and right, or above and below — must be 2 or 4, the only digits with two different doubling partners. Then read each boundary against any digit already beside it, and write what is left in as pencil marks:
| Digit | Across a black dot | Across a white dot | Across a blank boundary |
|---|---|---|---|
| 1 | 2 | — | 3, 4, 5, 6, 7, 8, 9 |
| 2 | 1, 4 | 3 | 5, 6, 7, 8, 9 |
| 3 | 6 | 2, 4 | 1, 5, 7, 8, 9 |
| 4 | 2, 8 | 3, 5 | 1, 6, 7, 9 |
| 5 | — | 4, 6 | 1, 2, 3, 7, 8, 9 |
| 6 | 3 | 5, 7 | 1, 2, 4, 8, 9 |
| 7 | — | 6, 8 | 1, 2, 3, 4, 5, 9 |
| 8 | 4 | 7, 9 | 1, 2, 3, 5, 6 |
| 9 | — | 8 | 1, 2, 3, 4, 5, 6, 7 |
What the neighbour can be, for each digit and each kind of boundary. A dash means that digit can never touch that dot. 1 and 2 are both one apart and double-and-half; this page always draws that pair black, which is why a white dot never has a 1 on it. The blank column is the one people forget, and it is the longest list only because it forbids the most specific things.
The Fill button does all of that for you. On a fresh Easy board it cuts the average square from 3.0 candidates to 1.8 and settles 24 squares outright; on Medium, 20. On Expert, where nothing is given, Fill can only use the shapes: it takes the average square from nine candidates to 7.3 and settles none, which is where the real work starts.
Are the blank boundaries really clues?
On the harder boards they are most of the puzzle. Every board this page deals has exactly one answer under all three rules. We re-counted each one with the blank rule switched off, so that only the drawn dots count:
On Easy and Medium the drawn dots and the givens are enough on their own — not one of 80 boards gained a second answer without the blanks, so there you can solve almost as if the blanks were not there. On Hard, 29 boards in 40 stop having a single answer, and on Expert it is more than half of those we could settle. Take the dots away altogether and even an Easy board has several answers; from Medium up it has hundreds. The empty grid of an Expert board is not a trick, either: across 200 random finished grids, the full dot layout with no digits at all pinned the grid it came from 187 times.
How do the four levels differ?
Here the dial is the number of given digits, and only that — with the blank rule in force every boundary’s state follows from the answer, so the dots cannot be thinned out to make a board harder. What changes is how much reading you have to do, and of what kind:
| Level | Given digits | Dots | Hardest step the board needs | Pencil marks per square after Fill | Squares Fill settles outright |
|---|---|---|---|---|---|
| Easy | 34 | 44 | Singles | 3.0 → 1.8 | 24 |
| Medium | 26 | 43 | A dot read | 4.0 → 2.4 | 20 |
| Hard | 8 | 43 | A blank read | 7.2 → 5.2 | 4 |
| Expert | 0 | 44 | A dot chain (31 of 40), or a classic technique beyond singles (9 of 40) | 9.0 → 7.3 | 0 |
40 boards per level from this page’s engine, fixed seed; medians. “Hardest step” is the highest rung of the engine’s own ladder a board needed: a dot read reads a drawn dot against a digit already in place, a blank read does the same with a boundary that has nothing on it, and a dot chain carries the relations between squares that are all still empty. Every board landed exactly on its level’s clue count and in the band it was dealt for. The pencil-mark column is before and after pressing Fill on a fresh board.
Each level asks for one new kind of reading. Easy is naked and hidden singles on candidates the dot shapes have already narrowed. Medium adds the dot read, Hard the blank read, and Expert the chain, where you follow relations through squares that are all still empty; nearly a quarter of Expert boards also want a classic technique such as locked candidates or a naked pair. The jump from Medium to Hard is the big one: 26 givens to 8.
The engine behind this page removes digits one at a time and keeps a removal only while the board still has exactly one answer under the dots and blanks you are shown; a search it cannot finish counts as a failure and the digit stays. We re-checked 160 boards with a separate solver that shares no code with it: every one had exactly one answer. Proving an 8-given or empty board takes real work, so the board is now built in the background: on a phone-speed processor the page used to freeze for up to half a second while a Hard or Expert board was made, and now it never stalls for more than a tenth of a second.
Which level should you play?
Start on Medium if you know sudoku. Twenty-six givens keep the grid familiar, and every board needs at least one dot read, so you learn the notation by using it. Easy is the right place if the dots are new; singles finish every board there.
Hard is where kropki becomes its own puzzle: eight digits, and on nearly three boards in four the blanks are what make the answer unique. Expert hands you an empty grid. Press Fill, find the squares boxed in by black dots, and follow the chains out from there; every Expert board has exactly one answer, and none needs guessing.
What goes wrong on a Kropki Sudoku
- Ignoring the blank boundaries. They are clues. On Hard and Expert, a solve that reads only the dots will find more than one way to finish most boards.
- Putting 1 and 2 across a white dot. That pair is always drawn black here, so a white dot never has a 1 on either side.
- Assuming three white dots in a row make four consecutive digits. They do when the four squares run in a straight line, because a line cannot repeat a digit. Round a corner, the first and third squares share no row or column and can be equal: 4, 5, 4 is a legal bend.
- Counting black dots around a square instead of along one line. A square needs to be 2 or 4 only when both black dots are on the same line. Black dots above and to the left can hold the same partner digit.
- Looking for a 5, 7 or 9 on a black dot. None of them can ever sit there.
Where does Kropki Sudoku come from?
Kropki was invented in 2005 by the Belarusian puzzle setter Vladimir Portugalov. It first ran in a Forsmarts contest as a puzzle with no instructions, took its name around the 2006 World Puzzle Championship, and has appeared at the championship many times since. The name is Belarusian — кропкі means “dots” — and not Polish, as this page used to say. The one thing puzzles differ on is the 1-and-2 pair: some let it take either dot, and this page always draws it black.
What do the buttons do?
Select a square, then a digit from 1 to 9; the page opens on Medium. Selecting a square also says which dots it touches, in the line under the grid, which is how a screen reader hears the dots. Under the grid:
- Notes turns pencil marks on and off. A correct digit clears itself from the notes of its row, column and box, and from its neighbours wherever a dot or a blank rules it out.
- Fill writes pencil marks into every empty square, already narrowed by the houses, the dot shapes and every boundary next to a placed digit. Undo takes the whole fill back in one step.
- Erase clears the selected square, and pressing the same digit twice does the same.
- Undo takes back the last change, including the notes a correct digit cleared, though not the Mistakes count.
- Hint finds the next logical step, gentlest first — a single, then a dot read, a blank read or a chain — makes it, and explains it in the line under the grid. Unlike most sudoku hints it never just reveals a random square.
- Solve asks first, then fills the grid and stops the timer.
- New Game deals another board at the level you are on; the level buttons deal one at theirs.
A wrong digit that breaks a dot or a blank is told which boundary it broke and why; a digit that breaks nothing but is still not the answer is told that too. Digits 1–9, the arrow keys, Backspace or Delete and Ctrl+Z all work from a keyboard, and the board is saved in this browser after every move.
Where to go next
Kropki is one of the boundary variants, where the clues sit between squares rather than in them. Consecutive Sudoku keeps only the one-apart rule, with a marker on the boundary, and the same rule that an unmarked boundary means the opposite. XV Sudoku swaps the dots for totals: an X where two squares add up to 10, a V where they add up to 5. Greater Than Sudoku puts an inequality sign between neighbouring squares instead, and Thermo Sudoku strings the same kind of ordering along a line.