Kropki Solving Techniques
Kropki is solved by reading the dots as a network of constraints. New here? Read the rules first.
1. Black-dot chains
Two black dots in a row mean three digits each double the last: 1-2-4 (or 2-4-8 where n is large). A black dot never touches an odd digit above 1 on its halved side: a 3, 5 or 7 can only be the smaller digit... and only when its double fits in the grid.
2. White-dot chains
A run of white dots is a run of consecutive digits going up or down, possibly turning around. A chain of four white dots in a 5×5 grid spans the whole 1 to 5, and the ends are 1 and 5.
3. The missing dots
No dot is a strong clue. A cell holding n (the largest digit) can only sit next to an undotted cell if that neighbour is not n−1 and not n/2. In a 4×4, a 4 next to an undotted cell forces that neighbour to 1 (not 3, not 2).
4. Odd digits and black dots
In a 7×7, a 5 or a 7 can never be on a black dot (no double fits, and they are not a double). In a 6×6 the same goes for 5; 3 can only pair with 6.
5. Notes for pairs
A white dot with one cell known leaves two candidates for the other. Pencil them in; the row and column usually remove one.
6. Corners and edges
Edge cells have fewer neighbours and so fewer dots to satisfy; corner cells with no dots at all are often the largest or an isolated digit like 5 in a 7×7.
Frequently asked questions
How do I start a Kropki puzzle with no given digits?
Look for long chains of black dots: three cells in a row joined by black dots must be 1-2-4 or 4-2-1 (or 2-4-8 in big grids). Then use white-dot chains, which are runs of consecutive digits, and the missing dots to pin them down.
Last updated 2026-10-21