Kropki Sudoku is a classic 9x9 Sudoku puzzle decorated with small dots on the borders between cells. Each dot encodes a relationship: a white dot means the two neighbouring digits are consecutive, and a black dot means one digit is double the other. Just as importantly, the absence of a dot is a clue too. The standard row, column, and box rules still apply, so Kropki combines familiar Sudoku logic with a satisfying layer of relationship deduction. The name comes from the Polish word "kropki," meaning "dots."
What is Kropki Sudoku?
Kropki Sudoku, sometimes called Dot Sudoku, is played on the same 9x9 grid as standard Sudoku. The grid may start with a few given digits, or none at all — the dots carry most of the information. Between some pairs of orthogonally adjacent cells (horizontal or vertical neighbours) you will find a small circle. There are two kinds:
- A white dot (○) marks a pair of consecutive digits — two numbers that differ by exactly 1, such as 4 and 5, or 8 and 9.
- A black dot (●) marks a pair in a 2:1 ratio — one digit is exactly double the other, such as 3 and 6, or 4 and 8.
The puzzle has a single unique solution that can always be reached through logic. Kropki is popular with solvers who enjoy spotting patterns and ratios rather than doing arithmetic sums, and it is a staple of the World Puzzle Championship.
Kropki Sudoku Rules
Kropki Sudoku has four rules. The first inherits all of classic Sudoku, and the next three govern the dots.
- Standard Sudoku rules: Each row, each column, and each 3x3 box must contain the digits 1 through 9 exactly once.
- White dot rule: Two cells separated by a white dot must contain consecutive digits (they differ by 1).
- Black dot rule: Two cells separated by a black dot must contain digits in a 2:1 ratio (one is double the other).
- No-dot rule: Two adjacent cells with no dot between them are neither consecutive nor in a 2:1 ratio. Every applicable relationship is marked, so a missing dot rules out both possibilities.
The Dot Pairs at a Glance
Knowing exactly which pairs each dot allows is the foundation of Kropki solving. The tables below list every valid pair on a 9x9 grid.
White Dot (Consecutive) Pairs
| Lower Digit | Consecutive Partner |
|---|---|
| 1 | 2 |
| 2 | 1 or 3 |
| 3 | 2 or 4 |
| 4 | 3 or 5 |
| 5 | 4 or 6 |
| 6 | 5 or 7 |
| 7 | 6 or 8 |
| 8 | 7 or 9 |
| 9 | 8 |
Black Dot (2:1 Ratio) Pairs
| Pair | Relationship |
|---|---|
| 1 and 2 | 2 = 1 × 2 |
| 2 and 4 | 4 = 2 × 2 |
| 3 and 6 | 6 = 3 × 2 |
| 4 and 8 | 8 = 4 × 2 |
Solving Strategies
1. Start with black-dot chains
Because only four pairs satisfy a black dot, chains of black dots are extremely restrictive. A run of three cells joined by black dots must follow a doubling sequence such as 1-2-4 or 2-4-8 (in some order). Identify these chains first; they often lock in several cells at once.
2. Treat "no dot" as information
The absence of a dot is one of the most powerful and most overlooked clues. If a cell is known to be 5 and a neighbour has no dot between them, that neighbour cannot be 4 or 6 (consecutive) and cannot be 10 (impossible) or anything that doubles/halves to 5 — effectively you remove 4 and 6 from the neighbour. Always check the borders that have no dot, not just the ones that do.
3. Use 1 and 9 as anchors
The digit 1 can only be consecutive with 2, and 9 can only be consecutive with 8. So a white dot on a cell limited to high values likely involves 8-9, and a white dot near low values likely involves 1-2. These anchors help you orient a chain of white dots.
4. Combine with standard Sudoku logic
Dot deductions narrow candidates, but you still finish the puzzle with classic techniques: naked singles, hidden singles, naked pairs, pointing pairs, and box-line reduction. Alternate between dot logic and standard elimination — each unlocks the other.
5. Map candidates around every dot
For each dot, pencil in the small set of pairs it allows, then cross off any that conflict with the row, column, or box. A white dot only permits consecutive pairs; a black dot only permits the four ratio pairs. Reducing each dot to its surviving options frequently reveals a forced placement.
Mini Kropki (6x6)
Mini Kropki brings the same white-dot and black-dot clues to a compact 6x6 grid using the digits 1 through 6 with 2x3 boxes. The rules are identical:
- Each row, column, and 2x3 box must contain 1-6 exactly once.
- A white dot marks consecutive digits (differ by 1).
- A black dot marks a 2:1 ratio.
- No dot means neither relationship holds.
On a 1-6 grid the only black-dot pairs are 1-2, 2-4, and 3-6, which makes black dots even more decisive than on the full grid. The digit 5 never touches a black dot at all. Mini Kropki is free in Sudoku - Brain Puzzles and is the ideal place to internalise dot logic — a typical puzzle takes just a few minutes, yet it exercises exactly the same reasoning as the 9x9 version.
Tips for Beginners
- Memorize the black-dot pairs. Only 1-2, 2-4, 3-6, and 4-8 work on a 9x9 grid. This single fact unlocks most black-dot deductions instantly.
- Always read the missing dots. Beginners focus on the dots that are present; experts also use the dots that are absent. The no-dot rule is half the puzzle.
- Use pencil marks. Note candidate pairs along each dot. Sudoku - Brain Puzzles updates pencil marks automatically as you place digits.
- Watch the 1-2 special case. Remember that 1-2 satisfies both a white and a black dot, so it can appear on either; don't rule it out from a white dot.
- Start with Mini Kropki. The 6x6 grid teaches the same logic far faster. Build confidence there before moving up to 9x9.
- Never guess. Every Kropki puzzle has a unique logical solution. If you stall, re-examine a black-dot chain or a no-dot border you skipped.
Kropki Sudoku rewards a sharp eye for relationships rather than arithmetic. Once you know the handful of dot pairs by heart and learn to read the empty borders, the dots turn from decoration into a precise solving path. Start small with Mini Kropki, and the full 9x9 puzzles will soon feel approachable.