Kropki Sudoku

White dots = consecutive digits; black dots = one digit is double the other

Easy

What is Kropki Sudoku?

Kropki Sudoku places dots between pairs of adjacent cells. A white dot means the two digits differ by exactly 1 (consecutive). A black dot means one digit is exactly double the other (e.g., 2 and 4, or 3 and 6). Where no dot appears, neither condition applies. Some Kropki puzzles use the 'negative constraint' where the absence of a dot guarantees neither condition holds, giving even more information.

At a Glance

Constraint type	Cell Relationships
Typical givens	18–24
Difficulty rating	★★★☆☆ 3/5
Avg. solve time — Easy	8 min
Avg. solve time — Medium	18 min
Avg. solve time — Hard	35 min
Avg. solve time — Expert	62 min

How to Solve Kropki Sudoku

Technique	What it does	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

Average Solve Times

Easy

8 min

Medium

18 min

Hard

35 min

Expert

62 min

Frequently Asked Questions

What exactly is the black dot relationship?

One digit is exactly double the other: (1,2), (2,4), (3,6), or (4,8). The pair (5,10) is invalid — 10 is out of range.

Can 5 appear in a black dot pair?

No. 5×2=10 is out of range, and 5÷2=2.5 is not an integer.