Solving kropki dots
A black dot joins a digit and its double: 1 and 2, 2 and 4, 3 and 6, or 4 and 8. A white dot joins two digits in a row. Put two dots together and the choices shrink fast.
Why it works
On the board above:
r4c3 sits between two black dots, so r4c2 and r4c4 are each half or double r4c3.
The three cells share row 4, so they are all different. The only doubling chains of three digits are 1, 2, 4 and 2, 4, 8.
The middle of either chain is 2 or 4. So r4c3 can only be 2 or 4.
More moves
Two dots, one cell
A cell on a white dot and a black dot must fit both. Its row or column can rule out the rest.
r6c1 is 2, and a white dot joins it to r6c2. So r6c2 is 1 or 3.
If r6c2 were 1, its black dot partner r6c3 would have to be 2. Row 6 already has a 2, in r6c1.
So r6c2 is 3, and r6c3 is 6.
How to use it
Know the black pairs
Only 1-2, 2-4, 3-6 and 4-8. 5, 7 and 9 never touch a black dot.
Follow chains
Two black dots in a row mean a doubling chain: 1-2-4 or 2-4-8. The middle is 2 or 4.
Mix the colors
A cell on a white dot and a black dot has to satisfy both. List its options for each and keep the overlap.
Check the houses
A dot’s partner may already sit in the same row, column or box. That cuts the options again.
Partners on a black dot
Practice
8 boards. Each has one right answer, checked by our solver.
Loading the boards…
Try it in a real puzzle
Each of these starts with dot logic.
Tips
White chains
Three cells joined by two white dots hold three digits in a run, like 4-5-6. The middle can’t be 1 or 9.
1 and 2 fit both
1 and 2 are both consecutive and double. A dot of either color can hold them.
Negative rules
If the rules say all dots are given, a pair with no dot is neither consecutive nor double. That rules out a lot.
Start at 3 and 6
3 and 6 have only each other on a black dot. They resolve fast.