Kropki Sudoku: rules, tips and free puzzles

Kropki sudoku puts small dots between neighbouring cells. The kropki sudoku rules are short: a white dot joins consecutive digits, a black dot joins digits where one is double the other.

The twist that makes kropki special is that every possible dot is shown. A missing dot is a clue too, and on a sparse grid it is often the strongest one you have.

Play Kropki Sudoku free

Opens in your browser. No sign-up, no ads.

The rules

Normal sudoku rules apply: 1 to 9 once in every row, column and 3×3 box.
A white dot between two cells: their digits are consecutive (they differ by exactly 1).
A black dot: one digit is exactly double the other.
Every possible dot is shown. Two neighbouring cells with no dot between them are neither consecutive nor in a 1:2 ratio.
A 1 and a 2 satisfy both relationships, so a 1–2 pair may carry either colour of dot.

How to start

Know the black-dot pairs

Only four pairs are in a 1:2 ratio: 1–2, 2–4, 3–6 and 4–8. So a cell with a black dot can only be 1, 2, 3, 4, 6 or 8; 5, 7 and 9 never touch a black dot.

Two black dots in a row. The middle digit must be double or half of two different digits, so it is 2 (with 1 and 4) or 4 (with 2 and 8).

Use the missing dots

Because all the dots are shown, every pair of neighbours without a dot is a negative clue. A 5 with no white dot to a neighbour means that neighbour isn’t 4 or 6. A 1 with no dot at all beside it means none of its neighbours is a 2, since 1–2 would need a dot of one colour or the other.

Follow white-dot chains

Three cells in a row joined by two white dots hold three consecutive digits with the middle cell in the middle, such as 4, 5, 6 in either direction. The middle cell can’t be 1 or 9.

Common mistakes

Ignoring the missing dots. In kropki they carry as much information as the dots themselves.
Assuming a 1–2 pair must have a white dot. It can have either.
Reading a black dot as a sum. It is always a ratio of exactly 1:2.

How the ranks scale difficulty

Every puzzle type on Hoot’s Puzzles comes in four ranks, named after the stages of a mathematical idea. Lemma is the warm-up: about a dozen more digits are given than logic strictly needs, and singles plus the variant’s own rule will carry you through. Proposition needs a real idea or two, typically pointing and claiming. Theorem is near-minimal, so almost every clue is load-bearing and you may need pairs. Conjecture gives the fewest digits and sparser rule clues: expect to think hard.

Every dot is drawn at every rank, because the missing-dot rule only works when the dots are complete. Rank changes how many digits you are given and how deep the logic goes.

There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.

Frequently asked questions

What are the kropki sudoku rules?

Normal sudoku, plus white dots join consecutive digits and black dots join digits in a 1:2 ratio. Every possible dot is shown, so neighbours without a dot are neither.

Which digits can sit on a black dot?

Only 1, 2, 3, 4, 6 and 8, from the pairs 1–2, 2–4, 3–6 and 4–8.

Why can a 1 and 2 have either dot?

They are consecutive and one is double the other, so both dots are true. Either colour may be shown.

Do dots apply diagonally?

No. Dots only sit between cells that share an edge.