Kropki Sudoku: Rules & How to Solve

Kropki is Polish for "dots". A white dot marks consecutive digits, a black dot a 1:2 ratio — here are the rules and the techniques that crack them.

The rule

What does “kropki” mean?

Kropki is simply the Polish word for dots — singular kropka. The variant takes its name from its only visual ingredient: the small circles printed on the edges between neighbouring cells. You will also see it written as Kropki dots, dot sudoku, or, when only the white dots are used, consecutive sudoku.

Dots sit on the edge between two orthogonally adjacent cells, never on a corner and never between diagonal neighbours.

How to solve kropki sudoku

Black dots are the strongest clue on the board

Only four pairs of digits from 1–9 stand in a 1:2 ratio: {1,2}, {2,4}, {3,6} and {4,8}. A single black dot therefore cuts both cells down to at most four candidates each, and immediately rules out 5, 7 and 9 from both — those three digits can never touch a black dot. Always resolve black dots before white ones.

Black-dot chains collapse fast

Two black dots in a row are far more restrictive than one. When the three cells also share a row, column or box — so no digit can repeat among them — the chain must read 1–2–4 or 2–4–8 in one direction or the other, and a four-cell chain is forced to 1–2–4–8. Where the chain wanders across regions it can fold back on itself (2–1–2, 4–8–4), so check the geometry before collapsing it.

Chains of white dots behave like renban lines

A run of white dots means every step moves by exactly one. When the cells of the run share a row, column or box, no digit can repeat, so the run becomes a block of consecutive digits — a short renban line without a fixed direction. Length then does the work: a five-cell run must span a window of five consecutive digits, so its lowest cell is at most 5 and its highest at least 5. A run that crosses between regions may zigzag instead (3–4–3–4), which constrains far less — confirm the cells share a region before treating a run as a renban.

Watch the 1–2 overlap

The pair {1,2} is the one case that satisfies both rules at once: 1 and 2 are consecutive and also differ by a factor of two. Either colour of dot can legally mark it, so a white dot never fully rules out {1,2} and neither does a black one. Setters usually state which convention they follow — read the rules before leaning on that elimination.

Use the absence of a dot carefully

When a puzzle states that all dots are given, an empty edge is itself a clue: those two neighbours are neither consecutive nor in a 1:2 ratio, which is often enough to break a pair. When the puzzle does not make that promise — as in VariantDoku's catalog — a missing dot means nothing at all, and treating it as information will send you down a false line.

Frequently asked questions

What does a white dot mean in Kropki sudoku?

The two cells it sits between hold consecutive digits, such as 4 and 5.

What does a black dot mean?

One cell is exactly double the other — {1,2}, {2,4}, {3,6}, or {4,8}.

What does kropki mean?

Kropki is Polish for “dots” (singular: kropka). The variant is named after the small dots printed between neighbouring cells, which are its only clue type.

Can 1 and 2 have a white or a black dot?

Both are valid. 1 and 2 are consecutive and also sit in a 1:2 ratio, so the pair satisfies either rule. Most setters state which dot they use for it, so check the puzzle's rules before ruling {1,2} out.

Which digits can never touch a black dot?

5, 7 and 9. The only 1:2 pairs available in 1–9 are {1,2}, {2,4}, {3,6} and {4,8}, and none of them contains 5, 7 or 9.

Are all Kropki dots always given?

Not necessarily. Many puzzles, including VariantDoku's, note when dots may be omitted, so don't assume a missing dot rules a relationship out unless stated.

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