Hard Kropki Sudoku

Black dots signal a doubling relationship; white dots mean consecutive digits. Unmarked pairs carry equally powerful information. Free online.

Easy Medium Hard Expert

Hard Kropki Sudoku. Dense dot networks and fewer givens — multi-dot chains restrict entire rows and boxes in unexpected ways.

2
6
8
7
1
4
2
5
9
4
5
3
1
6
3
4
5
5
6
3
9
2
2
8
1
8
7
Mistakes
0/3
Score
-
Time
00:00
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Progress0%

What is Kropki Sudoku?

Difficulty
★★★☆☆
3/5
Constraint Type
Cell Relationships
Typical Givens
18–24
Avg. Solve (Hard)
35 min

Solving Techniques for Hard Level

Technique Description Level
White Dot Pairs White dots mark consecutive digit pairs: (1,2), (2,3), …, (8,9). Cross-reference with row and column constraints. Beginner
Black Dot Pairs Black dots mark double-relationship pairs: (1,2), (2,4), (3,6), (4,8). Only these four combinations are valid. Beginner
Negative Constraint No dot between two cells means they are neither consecutive nor in a 1:2 ratio — a powerful elimination on every unmarked adjacent pair. Intermediate
Dot Chain Analysis A sequence of connected dots forms a digit-value chain. Enumerate valid digit sequences for the entire chain simultaneously. Advanced
1-2-4 and 2-4-8 Chains A black dot adjacent to a white dot often forces the common 1→2→4 or 2→4→8 chain. Intermediate

Ready for the ultimate test? Try Expert Kropki Sudoku — the hardest puzzles on Sudoku.by.

Average Solve Time by Difficulty

Easy
8 min
Medium
18 min
Hard
35 min
Expert
62 min
Want a full walkthrough of rules, strategies, and solving steps? How to Play Kropki Sudoku →

Frequently Asked Questions — Hard Kropki Sudoku

What makes hard Kropki Sudoku difficult?
Hard puzzles have fewer pre-filled digits and a denser, more interconnected network of dots. You need to chain dot constraints across multiple adjacent cells — for instance, a sequence of three cells linked by two dots creates a highly restricted triple that eliminates candidates from many other cells in the row or box.
What is a dot chain in Kropki Sudoku?
A dot chain is a sequence of cells connected by consecutive dots, such as A–(black)–B–(white)–C. The chain constrains all three cells simultaneously. For example, if A–(black)–B means {1,2} or {2,4} pairs, and B–(white)–C means B and C differ by 1, the combined constraint on B's value is very tight. Chains of length 3 or 4 are the core technique at hard level.
Should I use pencil marks for hard Kropki Sudoku?
Yes — hard puzzles require pencil marks for every empty cell. Write candidate digits and also annotate which pairs are still valid for each dot. When you eliminate a candidate from one cell, immediately propagate that through all dots touching that cell. Falling behind on propagation is the main reason players stall on hard Kropki.
How does a 1:2 chain work in practice?
A 1:2 chain involves multiple black dots: if A–(black)–B–(black)–C, then the possible values cascade as 1–2–4 or 2–4–8 or 4–8–? (impossible). This limits A, B, and C to at most three specific triples. Combined with row/column/box constraints, this usually pins down one of the three cells' values exactly.
How long does a hard Kropki Sudoku take to solve?
Expect 35–60 minutes for hard Kropki Sudoku. The toughest moments are usually chains of three or more dots where you need to enumerate all valid triples and cross-reference them against the box. Once you find a valid triple that is unique, the rest of the grid typically unlocks in a cascade.

More questions? See the full Kropki Sudoku guide.