Play Kropki Sudoku Online

Kropki Sudoku: The Dot-Based Logic Puzzle

Kropki Sudoku (also known as Kropki Dots or Dot Sudoku) is an elegant variant that enriches the classic Sudoku experience with a simple yet powerful clue system. Between adjacent cells, black and white dots indicate mathematical relationships — consecutive digits or a 1:2 ratio. The twist? If there is no dot between two neighbours, neither relationship holds, making every empty gap a clue in itself.

🤔 What Is Kropki Sudoku?

A Kropki Sudoku puzzle uses a standard 9×9 grid divided into nine 3×3 boxes, just like regular Sudoku. The key difference is the dots placed on cell boundaries:

• White dot (○) — the two adjacent digits are consecutive (they differ by exactly 1). For example, 3 and 4, or 7 and 8.
• Black dot (●) — one digit is exactly double the other. For example, 2 and 4, or 3 and 6.
• No dot — the two digits are neither consecutive nor in a 1:2 ratio. This negative constraint is just as important as the dots themselves.

The word Kropki is Polish for "dots," giving the puzzle its distinctive name.

📋 Rules of Kropki Sudoku

Kropki Sudoku combines standard Sudoku rules with dot constraints:

1. Standard Sudoku rules — Every row, column, and 3×3 box must contain the digits 1–9 exactly once.
2. White dot rule — If a white dot appears between two cells, the digits in those cells must be consecutive (differ by 1).
3. Black dot rule — If a black dot appears between two cells, one digit must be exactly double the other.
4. Negative constraint — If there is no dot between two adjacent cells, the digits must be neither consecutive nor in a 1:2 ratio.

Note: The pair 1 and 2 satisfies both the consecutive and the double condition. In such cases a black dot is shown (some variants use either; our version follows the standard convention of showing black when both apply).

⭐ Difficulty Levels Explained

The dial here is the given count, and — unusually — there is no second one. Because the negative constraint is in force, every boundary's state is implied by the solution, so we cannot thin the dots out to make a puzzle harder. Leaving a dot off would not make it harder; it would make it wrong. Every board below is proved to have exactly one answer under the dots you are actually shown.

• Easy — 34 given digits. Plenty of anchor points, and singles will carry you through once the dot shapes have narrowed the candidates.
• Medium — 26 givens. The level where you start reading a dot: a drawn dot against a digit already on the board.
• Hard — 8 givens. The empty boundaries now do most of the work. Expect to lean on the negative constraint on nearly every board.
• Expert — no given digits at all. The dots and the gaps between them are the entire puzzle. Solving means carrying relations between cells that are all still empty.

🧠 Essential Kropki Sudoku Strategies

Mastering Kropki Sudoku takes practice. Here are the key techniques:

1. Dot Candidate Analysis

For each dot, list the possible digit pairs. A white dot allows pairs like {1,2}, {2,3}, {3,4}, {4,5}, {5,6}, {6,7}, {7,8}, {8,9}. A black dot allows {1,2}, {2,4}, {3,6}, {4,8}. Cross-reference with row/column/box constraints to narrow possibilities.

2. The Negative Constraint

Where there is no dot, eliminate all consecutive and double pairs from consideration. For instance, if a cell contains 4, its dot-free neighbour cannot be 3, 5, 2, or 8. This constraint is surprisingly restrictive and often resolves cells directly.

3. Chain Deduction Along Dots

When multiple dots form a chain (e.g., white-white-white), the digits must form a consecutive sequence. A chain of three white dots means four cells contain four consecutive numbers. This severely limits the possibilities and often resolves the entire chain.

4. Parity Reasoning

White dots alternate between odd and even digits (since consecutive numbers always switch parity). If one cell on a white dot is even, its partner must be odd. Chain this through connected white dots to lock down entire groups.

5. Standard Sudoku Techniques

All classic Sudoku strategies — naked singles, hidden singles, pointing pairs, box-line reduction, X-Wing, and more — still apply. Combine them with dot logic for powerful multi-step deductions.

📊 Key Dot Pair Reference

Memorise these allowed digit pairs to speed up your solving:

• White dot pairs: {1,2}, {2,3}, {3,4}, {4,5}, {5,6}, {6,7}, {7,8}, {8,9}
• Black dot pairs: {1,2}, {2,4}, {3,6}, {4,8}
• Both conditions (1 & 2): The pair {1,2} satisfies both consecutive and double — shown as a black dot
• Digits that can never touch a black dot: 5, 7, 9
• Max digit on a black dot: 8 (since 9 has no double ≤ 9)

🆚 Kropki Sudoku vs. Regular Sudoku

• Clue type: Regular Sudoku uses given digits. Kropki Sudoku uses dots (and given digits on easier levels).
• Negative information: In Sudoku, unmarked cells carry no extra info. In Kropki, the absence of a dot is a powerful clue.
• Mathematical reasoning: Kropki adds ratio and consecutive logic on top of standard elimination.
• Visual style: The black and white dots give the grid a distinctive, visually appealing look.

📜 History of Kropki Sudoku

Kropki Sudoku emerged from the competitive puzzle community. The variant gained international recognition through the World Puzzle Federation (WPF) instruction booklets in the early 2010s. The name "Kropki" (Polish for "dots") reflects the puzzle's Central European origins. It has since become a regular fixture at the World Sudoku Championship and is widely featured in online puzzle competitions and apps.

💪 Benefits of Playing Kropki Sudoku

• Strengthens logical reasoning — juggling positive and negative constraints sharpens deduction skills.
• Improves number sense — constant work with consecutive and ratio relationships builds mathematical intuition.
• Enhances pattern recognition — spotting dot chains and parity patterns trains your brain to see structure.
• Deeply satisfying — cracking a tricky dot-chain deduction delivers an exceptional "aha" moment.

🎮 More Sudoku Variants to Explore

• Classic 9×9 Sudoku — The original puzzle. Start here if you're new.
• Killer Sudoku — Cage sums replace given digits for an arithmetic twist.
• Thermo Sudoku — Digits must increase along thermometer lines.
• Arrow Sudoku — Digits along arrows sum to the circle digit.