Kropki Sudoku: How to Solve with Black and White Dots
A Kropki Sudoku is an ordinary Sudoku with small dots on the edges between neighbouring cells. A white dot means the two digits differ by exactly one. A black dot means one of them is exactly double the other.
The dots sit between cells rather than in them, which is the whole reason the variant plays differently. Every other clue you are used to lives inside a cell and tells you about that cell. A dot tells you about a relationship, and it gives you nothing at all until you know one of its two ends.
One thing to settle before you start, because it changes how you solve: find out whether the puzzle draws every dot that applies. Some do, and then a bare edge is a clue in its own right. Most do not, and then a bare edge means only that you have not been told anything. The section below is about the first kind, and this page’s boards are that kind.
The rules that do not change
Every row, column and 3x3 box still holds each of 1 to 9 exactly once, and there is still one solution reachable without guessing. The rules and the step-by-step guide cover the base game, and naked singles and hidden singles is the technique the deductions below are made of.
A black dot is worth much more than a white one
Count what each dot leaves open.
A white dot has eight possible pairs — 1 and 2, 2 and 3, all the way to 8 and 9. Knowing one end tells you the other is one of two digits, and only at the very edges, next to a 1 or a 9, does it tell you the answer outright.
A black dot has four. One and 2, 2 and 4, 3 and 6, 4 and 8, and nothing else, because doubling anything from 5 upwards leaves the grid. So a black dot cuts much harder: three of the nine digits — 5, 7 and 9 — cannot sit beside a black dot at all. If you see a black dot, look there first.
That asymmetry is the single most useful thing to carry into a Kropki puzzle. The white dots are plentiful and weak; the black dots are scarce and strong.
The dot that is not there
Here is the part people miss on a completely marked puzzle, and it is where most of the solving actually happens on one.
Complete marking means the setter has drawn every dot that applies. On such a board, if two neighbouring cells have no dot between them, that is a statement: their digits are not consecutive and not in a two-to-one ratio. An empty edge becomes a clue with the same standing as a marked one, and there are far more empty edges than marked ones.
This is a convention, not part of the definition of Kropki, and a puzzle using it says so. Where the rules are silent, assume it is not in force — the dots you can see are then all you have. The board below is completely marked, which is what licenses the deduction that follows.
Take r2c4, the marked cell. Ordinary Sudoku gets it down to a 1 or an 8 and stops.
Look directly above it. r1c4 holds a 9, and the edge between them is bare. If r2c4 were an 8, then 8 and 9 would be consecutive and that edge would have had to carry a white dot. It does not. The 8 is impossible, and r2c4 is a 1.
Nothing was added to the board to make that work. The deduction runs entirely on a dot that was never drawn.
Once the 1 is placed, the black dot below it does the rest: the only digit that pairs with 1 across a black dot is 2, since halving a 1 leaves the grid altogether. r3c4 is a 2.
Why the dots do most of the work
On the completely marked board on this page, ordinary Sudoku technique — naked singles and hidden singles, using rows, columns and boxes only — proves four cells. Add the dots, positive and negative together, and the same technique proves twenty five.
That ratio is what a Kropki puzzle is. The grid is deliberately left thin, because the dots are carrying the puzzle.
Try it
In Sudoku Fog, Kropki is one of the variants you unlock rather than start with. Unlock points come from wins on Hard or above and from the daily challenge — one each, once a day — with bonus points at streak milestones.
Four ways this goes wrong
Reading a dot before you know either end. A dot constrains a pair. Until one of the two cells has a digit, it narrows nothing, and time spent staring at it is time lost. Work from the cells you have outward.
Forgetting that 1 and 2 satisfy both dots. They differ by one and one is double the other, so that pair can legitimately appear under either colour. It is the one place where a black dot does not exclude what you would expect.
Ignoring the empty edges on a completely marked puzzle. There they outnumber the dots several times over and every one of them is a real constraint. If such a puzzle feels stuck, look at what is not marked around the cell you are working on.
Assuming the negative rule always applies. It holds only where the puzzle marks every dot that applies, and that has to be stated. Read it into a puzzle that draws selected dots and you will produce contradictions that look like your own arithmetic mistakes. Sudoku Fog’s Kropki boards are of that second kind: they mark only some of the pairs, so a bare edge in the app tells you nothing and every deduction has to start from a dot you can see.
Where to go next
Thermometer is the other variant whose clues are about relationships rather than single cells, and it rewards the opposite instinct: there, the extremes are where the information is. King’s Move, Anti-Knight and X-Sudoku covers the three variants that add excluded cells instead of relationships.
If you want the version where most of the grid is hidden while you solve, fog of war Sudoku is that. Everything published so far is on the Learn index.
Sudoku Fog is on Android — What Sudoku Fog is, Get it on Google Play.