Kropki Sudoku

📱 Kids Sudoku: 4x4, 6x6, 9x9 Puzzles! No Ads!

Fun and colorful Sudoku puzzles designed for kids to boost logic and thinking skills. No ads, 100% kid-safe.

More Sudokus

What Is Kropki Sudoku?

Kropki Sudoku is a Sudoku variant that adds dot-based relationships between adjacent cells.

In addition to standard Sudoku rules:

• Each row contains every digit exactly once
• Each column contains every digit exactly once
• Each box contains every digit exactly once

Kropki Sudoku introduces two types of dots between orthogonally adjacent cells:

• Black dot: one digit is exactly double the other
• White dot: the two digits differ by exactly 1

If no dot appears between two adjacent cells, neither relationship is allowed. This "absence of dots" rule is just as important as the dots themselves.

Kropki Sudoku is commonly played in 6×6, 8×8, and 9×9 grids, each offering a different balance between arithmetic logic and spatial reasoning.

Understanding the Dots

Black Dots: The 2× Relationship

A black dot means one number is exactly twice the other:

• (1,2)
• (2,4)
• (3,6)
• (4,8)

Important detail:

• Order doesn't matter — either cell may hold the larger number
• Digits still cannot repeat in the same row, column, or box

White Dots: The ±1 Relationship

A white dot means the two numbers differ by exactly 1.

• (1,2)
• (2,3)
• (3,4)
• (4,5)
• (5,6)
• (6,7)
• (7,8)
• (8,9)

This is a much looser constraint than a black dot — but becomes powerful when combined with Sudoku restrictions.

No Dot Means "None of the Above"

If two adjacent cells do not have a dot:

• They are not consecutive
• They are not double / half

Many beginners miss this.

Experienced solvers rely on it constantly.

Tips for Solving Kropki Sudoku

Kropki Sudoku is very different from standard Sudoku, especially when the puzzle starts with few or even no given numbers.

In many puzzles, the dots — not the digits — are the real starting point.

Strong Kropki solving comes from learning how dot patterns forbid numbers before any placement is made. Each technique below tells you exactly whether it allows you to fix a value or eliminate candidates.

1. Black Dot Exclusion Rule

A black dot means one number is exactly double the other. Because of this, certain digits can never appear in any cell touching a black dot.

Examples:

• 6×6: Cells touching a black dot cannot be 5
• 9×9: Cells touching a black dot cannot be 5, 7, or 9

This does not place a number — but it removes high-impact candidates globally.

Experienced solvers apply this rule to every black dot before thinking about rows or boxes.

2. Multi-Dot ≠ 1 Rule

The digit 1 has very limited options:

• It can only form a black dot with 2
• It can only form a white dot with 2

So if a cell touches two or more dots, and those dots must all be satisfied in the same row, column, or region, then that cell cannot be 1.

This rule is extremely useful in dense dot areas and often removes 1 from multiple cells at once.

3. Double Black Chain

When three consecutive cells in the same row, column, or region are connected by two black dots:

the possibilities collapse immediately.

• 6×6: The only valid set is {1, 2, 4}, the middle cell can only be 2.
• 9×9: The only valid set is {1, 2, 4, 8}, the middle cell can only be 2 or 4.

This pattern rarely gives a fixed digit immediately, but it locks the structure so tightly that other placements soon follow.

4. Black–White–Black Sandwich

When four cells form with two black dots on both sides and a white dot in the middle.

the middle (white-dot) cell must contain 3.

Why?

• Black dot requires a 1:2 relationship
• White dot requires a difference of exactly 1
• Only combinations involving 3 satisfy both simultaneously

So, middle cell must include 3, possible pairs are: {2, 3} or {3, 4}.

This is not a guess — it is forced by arithmetic.

5. Closed Dot Loop (Ring Pressure)

When dots form a closed loop, constraints stack.

A common example:

• One black dot + three white dots
• All within the same box

In this structure, the black-dot pair must be 3 and 6.

No other values can satisfy all constraints around the ring.

This is a powerful mid-game technique and often gives direct placements, not just eliminations.

6. White Chain Pressure

Multiple white dots in the same row, column, or region create a strong bias toward consecutive numbers.

This usually means:

• Extreme values (1 and 9) become impossible
• Mid-range values cluster together

White-dot chains rarely give immediate placements, but they sharply narrow the search space.

7. No-Dot Rule

If two adjacent cells do not have a dot, then:

• They are not consecutive
• They are not in a 1:2 relationship

Advanced solvers actively look for missing dots, not just visible ones.

Kropki Sudoku FAQs

• Why does Kropki Sudoku often start with an empty grid?

  This is intentional.

  Many Kropki Sudoku puzzles are designed with no given digits at all, because the dots themselves are the information.

  If numbers were filled in at the start, most dot logic would collapse too quickly.

  A good Kropki puzzle forces you to extract meaning from:

  • where dots exist
  • where dots are missing
  • how dots cluster in rows, columns, or boxes

  If the grid looks empty but has many dots, it is not actually empty.

• What does "no dot" between two cells really mean?

  "No dot" is not neutral — it is a strong restriction.

  If two adjacent cells have no dot between them:

  • they cannot be consecutive (no white dot)
  • they cannot be double (no black dot)

  In 6×6 and 9×9 grids, this often removes more candidates than a dot would.

  Experienced solvers actively search for no-dot edges, especially:

  • inside boxes
  • along long chains of dotted cells

• Why do some digits almost never appear next to black dots?

  Because black dots impose extreme restrictions.

  A black dot means one number is exactly double the other. That immediately excludes:

  • 5, 7 and 9 in most grid sizes
  • 5 entirely in 6×6
  • many mid-range digits when dots cluster

  If a cell touches multiple black dots, the list of possible digits can shrink to one or two very quickly.

• How can I make progress when no single digit can be placed?

  Kropki Sudoku is often about candidate certainty, not placement.

  Many core moves:

  • do not place a number
  • only confirm which digits must belong to a small group

  For example:

  • three cells linked by black dots may be forced to use {1,2,4,8}
  • a boxed structure may force outer cells to avoid those digits

  These deductions quietly unlock the grid later.

• Why do black dots feel stronger than white dots?

  Because doubling is rarer than being consecutive.

  White dots allow many pairs: (1,2), (2,3), (3,4), …

  Black dots allow very few: (1,2), (2,4), (3,6), (4,8)

  When black dots cluster:

  • they create fixed digit sets
  • they often determine the scale of numbers in that area

  White dots usually fine-tune positions after black dot logic is applied.

• Can Kropki Sudoku be solved without guessing?

  Absolutely.

  A well-designed Kropki Sudoku is fully logical.

  If guessing feels necessary, it usually means:

  • a dot pattern was treated locally instead of globally
  • a "no dot" restriction was overlooked
  • a forced digit set was found but not applied elsewhere

  Stepping back and re-evaluating dot structures almost always reveals the missed logic.

• What is the single biggest mistake new players make?

  Treating dots as hints instead of constraints.

  New players often ask: "What does this dot allow?"

  Experienced solvers ask: "What does this dot forbid — especially when combined with others?"

  That mental shift is what makes Kropki Sudoku solvable from an empty grid.

• Is Kropki Sudoku closer to math or pure logic?

  Pure logic.

  No arithmetic is required beyond:

  • doubling
  • difference of one

  The challenge is not calculation — it is pattern pressure: how restrictions accumulate until only one outcome remains.