How to Solve Kropki Sudoku

A kropki sudoku often starts with an empty grid. The black and white dots between the cells carry all the information, together with the rule that every pair without a dot is neither consecutive nor double. This page shows how to turn the dots into digits, in the order that works best. The rules are explained in What is kropki sudoku?.

1. Know what each dot allows

A white dot joins two consecutive digits: 1–2, 2–3, 3–4 and so on up to 8–9. A black dot joins a digit and its double: 1–2, 2–4, 3–6 and 4–8 only. These four pairs are the whole list, so a cell next to a black dot can never be 5, 7 or 9. That rules out three of the nine digits before you have placed anything.

The pair 1–2 fits both dots. The setter chooses which colour to show, so a white dot can hide a 1–2 and so can a black dot.

2. Start with black-dot chains

Two black dots in a row, joining three cells, leave only two possibilities: 1–2–4 or 2–4–8, in either direction. The middle cell is 2 or 4. Three black dots joining four cells force 1–2–4–8, in one of two directions, so the two middle cells are 2 and 4.

Chains are the best starting point of any kropki. Find the longest black chain, write its two options as pencil marks, and look for a row, column or box that rules one of them out. Often a neighbouring dot does it: a white dot leaving the chain from a cell that must be 8 forces 7 or 9 next to it, and if that cell has a black dot on its other side, 7 and 9 are impossible, so the 8 cannot be there.

3. Then the white-dot runs

A run of white dots is a sequence of consecutive digits, going up or going down. A run of four cells is one of 1-2-3-4, 2-3-4-5, up to 6-7-8-9, or the same sequences reversed. The longer the run, the fewer the options. A run of five cells inside one row has only ten possibilities, and the row's other dots usually remove most of them.

Write the lowest and highest digit a run can reach. If a run of four cells sits in a row that already holds a 1 and a 9 elsewhere, the run is 2-3-4-5 up to 5-6-7-8, in either direction, so its end cells are between 2 and 8.

4. Use the no-dot rule all the time

Every pair of neighbours that qualifies for a dot has one. So a border without a dot tells you the two digits are not consecutive and not double. This is the rule beginners forget, and it is the rule that solves most of the grid.

As soon as you place a digit, look at its neighbours without a dot. Place a 4, and every dotless neighbour cannot be 3, 5, 2 or 8. Place a 1, and every dotless neighbour cannot be 2. Place a 9, and every dotless neighbour cannot be 8.

The no-dot rule also works before any digit is placed. A cell with no dots on any side cannot be next to a consecutive digit in any direction. If that cell's row already forces it to be 1 or 2, then its neighbours cannot be 1, 2 or 3 and, for the 1, not 2 either. Chains of such cells sometimes fix a whole region.

5. Combine the dots with sudoku logic

Once the chains have given you a few digits, treat the grid as a normal sudoku with extra information. Keep the pencil marks of the chain cells, because each has only two or three candidates. These cells form pairs and triples that remove candidates from their row, column and box.

A useful pattern is a black dot whose two cells both lie in one box. The box must contain both digits of a doubling pair. If the box already holds a 2, the pair cannot be 1–2 or 2–4, so it is 3–6 or 4–8.

Another pattern is a white dot on the border between two boxes. The two cells hold consecutive digits, and each digit is then barred from the other box in that row or column.

6. A solving order that works

  1. Mark all black chains with their options, and note that any cell on a black dot is not 5, 7 or 9.
  2. Mark the long white runs with their lowest and highest possible digit.
  3. Look for a cell where a chain option clashes with a row, column or box, and remove that option.
  4. Place the digits that are now forced. After each one, apply the no-dot rule to its neighbours.
  5. Go back to the chains. One placed digit usually settles the direction of a chain, and that gives several digits at once.

7. Mistakes to avoid

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