Kropki Sudoku Rules: White and Black Dots

By Andrew Boyko, developer of CageDoku · Updated October 23, 2026

Kropki Sudoku adds dots between neighbouring cells. A white dot means the two digits differ by exactly 1; a black dot means one digit is exactly double the other. The dots make every cell feel connected to its neighbours.

The rules

1. Every row, column and 3×3 box contains 1 to 9 once.
2. White dot: the two cells hold consecutive digits (for example 3 and 4).
3. Black dot: one cell holds exactly double the other (for example 3 and 6).
4. In puzzles with negative constraints, any pair of neighbouring cells without a dot must be neither consecutive nor in a 1:2 ratio.

The only black-dot pairs

There are exactly four pairs for a black dot: 1-2, 2-4, 3-6 and 4-8. That means a cell on a black dot can never be 5, 7 or 9. The pair 1-2 satisfies both dots, so a cell pair that carries both a white and a black dot must be 1 and 2.

White-dot chains

If three cells in a row are linked by two white dots, the digits are three consecutive numbers: the middle cell is one more than one neighbour and one less than the other. The middle cell therefore cannot be 1 or 9.

Technique: use the no-dot rule

In negative-constraint puzzles, a missing dot is information. Two adjacent cells without a dot can be neither consecutive (1-2, 2-3 …) nor in a 1:2 ratio (1-2, 2-4, 3-6, 4-8). This removes many candidates at once.

Practice

Try Kropki puzzles in CageDoku: hints name the technique and highlight the dots involved. Play a puzzle in your browser.

Frequently asked questions

Which pairs does a black dot allow?

1-2, 2-4, 3-6 and 4-8.

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

Yes: they are consecutive and double, so both dots can apply to the same pair.

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