Puzzle guide

How to solve Kropki

Kropki is a number grid with dots between the cells. A white dot joins two numbers one apart, a black dot a number and its double. Every dot is shown, so where there is no dot, that tells you something too. Most boards start with no numbers at all: the dots are the whole puzzle.

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The rules

Fill every cell with a number from 1 up to the size of the puzzle. Every number appears exactly once in each row and each column.
A white dot joins two numbers one apart, like 3 and 4.
A black dot joins a number and its double, like 2 and 4. Between a 1 and a 2 either dot may stand.
Every dot is shown: no dot between two cells means neither.

A solved puzzle

A solved 4×4 puzzle. In the top row, 3 and 2 have a white dot, 2 and 4 a black one, and 1 and 3 have none, because they are neither one apart nor double.

Your first moves

Look at black dots: on a 4×4 puzzle only 1 and 2, or 2 and 4, can stand on one. A 3 never touches a black dot.
Look at missing dots too: two cells with no dot between them can never be one apart or double.
Write notes for the numbers each cell can still hold.
Fill a row or column as soon as only one number is left for a cell.

Techniques, step by step

The last number in a row

This row already holds 4, 2 and 3. Every number appears once in a row, so the last cell is the missing 1. The same works for columns.

4, 2 and 3 are there, so the last cell is 1.

A black dot: doubles

A black dot means one number is twice the other. Next to a 4, the only number that fits is its half, 2, because 8 is too big. Black dots are rare and very strong: on a small puzzle, a cell beside one can only be 1, 2 or 4.

4 with a black dot: the neighbour is 2.

A white dot: one apart

A white dot means the two numbers differ by one. Next to a 3 that leaves only 2 or 4. Chains of white dots are even better: three cells joined by white dots hold three numbers in a row, like 2 3 4.

3 with a white dot: the neighbour is 2 or 4.

No dot is a clue too

This is the rule people forget. There is no dot between these cells, so their numbers are neither one apart nor double. On a 4×4 puzzle, next to a 3, that rules out 2 and 4. The 3 itself cannot repeat, so the cell is a 1. On bigger boards, missing dots are where the hard steps hide.

No dot next to a 3: not 2, not 4, so 1.

Pairs and hidden pairs

If two cells in a row can only hold the same two numbers, those numbers are used up there: cross them out of every other cell in the row. The other way round, if two numbers can only go into the same two cells of a row, those cells hold nothing else. The hard boards need both.

Common mistakes

Ignoring the missing dots. No dot is a clue: those two numbers are neither one apart nor double.
Mixing up the colours. White means one apart, black means double.
Forgetting 1 and 2. They are one apart and double at once, so either dot may stand between them.
Checking only the row. Every number must also be new in its column.

Questions and answers

What is Kropki?

A number puzzle on a square grid where each row and column holds every number once. White dots between cells join numbers one apart, black dots join a number and its double, and no dot means neither.

What does Kropki mean?

It is Polish for "dots".

Are all dots shown?

Yes. Every pair of neighbours that deserves a dot has one, so a missing dot is information too.

Why are there almost no numbers on the board?

Because the dots are enough. Most boards start with no printed number at all, or just one or two.

How big are the puzzles?

From 6×6 on Easy to 9×9 on Expert.

Ready to play?

Read the black dots, follow the white chains, and never forget that a missing dot is a clue. That is how every Kropki grid fills up.