1 of 3
Tap every digit that could sit on this black dot.
Select all that apply
Level 2 · Sums and pairs
How to solve Kropki sudoku
A white dot joins digits that are one apart. A black dot joins digits where one is double the other.
Kropki sudoku adds small dots on the borders between cells. A white dot means the two digits are one apart, like 4 and 5. A black dot means one digit is double the other, like 3 and 6.
The dots cut each cell down to a handful of options. Check the last line of the rules too. If all dots are given, every pair without a dot is neither consecutive nor in a 1:2 ratio, so the blank borders carry clues as well.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- White dot: consecutive digits. Black dot: one digit is double the other.
- Digits separated by a black dot are in a 1:2 ratio.
- Not all dots are given. / All dots are given.
Worth memorizing
- 5, 7 and 9 can never sit on a black dot. Their doubles are over 9 and their halves aren’t whole.
- 1 and 2 fit both colors: they’re consecutive and one is double the other.
- Watch the last line of the rules. “All dots are given” means every pair without a dot is neither consecutive nor 1:2.
Solving tips
5, 7, 9 skip black dots
Their doubles are over 9 and their halves are not whole numbers. A cell on a black dot can only be 1, 2, 3, 4, 6 or 8.
Black dot pairs are few
The only 1:2 pairs are 1 and 2, 2 and 4, 3 and 6, and 4 and 8. A 3 on a black dot must pair with a 6, and an 8 must pair with a 4.
Read black dot chains
Three cells in a straight line joined by two black dots must be 1, 2, 4 or 2, 4, 8, with the 2 or the 4 in the middle. No other middle digit has two different partners.
Use the all-dots rule
If all dots are given, neighbors without a dot are not consecutive and not in a 1:2 ratio. An 8 with a blank border beside it can't have a 4, 7 or 9 across that border.
A worked example: a white dot and a black dot in a row
Three cells sit side by side in a row. Call them A, B and C from left to right. A white dot sits between A and B, and a black dot sits between B and C. The row already has a 4 and a 6 in other cells.
- B is on a black dot, so it can't be 5, 7 or 9. It can be 1, 2, 3, 4, 6 or 8.
- The row's 4 and 6 remove those from B. That leaves 1, 2, 3 or 8.
- Test each one. A 1 at B would force both A and C to be 2, and they share a row. With 3 at B, C must be 6, which is already in the row. With 8 at B, C must be 4, which is also in the row. So B is 2.
- C is on a black dot with 2, so it is 1 or 4. The 4 is taken, so C is 1.
- A is on a white dot with 2, so it is 1 or 3. The 1 is now at C, so A is 3. The three cells read 3, 2, 1.
Practice
2 of 3
What goes next to the 9?
3 of 3
What goes next to the 3?
Reference table
| Digit | White dot partners | Black dot partners | Can't sit across a blank border |
|---|---|---|---|
| 1 | 2 | 2 | 2 |
| 2 | 1 or 3 | 1 or 4 | 1, 3 or 4 |
| 3 | 2 or 4 | 6 | 2, 4 or 6 |
| 4 | 3 or 5 | 2 or 8 | 2, 3, 5 or 8 |
| 5 | 4 or 6 | none | 4 or 6 |
| 6 | 5 or 7 | 3 | 3, 5 or 7 |
| 7 | 6 or 8 | none | 6 or 8 |
| 8 | 7 or 9 | 4 | 4, 7 or 9 |
| 9 | 8 | none | 8 |
Going deeper
Chains of white dots
When white dots join several cells in a straight line, and the cells share a row or column, the digits run in order. A chain of four cells could be 3, 4, 5, 6 or 6, 5, 4, 3, but never 4, 3, 4, 5. The middle cells can't turn back, since that would repeat a digit.
So a straight white chain of n cells behaves like a thermo that can run either way, with steps of exactly 1. A chain of 4 has ends that are exactly 3 apart. Once you know one digit and which way the chain runs, you know the rest.
Mixed chains are tighter still. A white dot next to a black dot, both on the same middle cell, gives that cell two different partners from two short lists. The worked example above shows how fast that can finish three cells when a couple of digits are already placed in the row.
Blank borders when all dots are given
If the rules say all dots are given, every border without a dot is a clue. The two cells can't be consecutive and can't be in a 1:2 ratio. The last column of the table shows what each digit rules out across a blank border.
This hits middle digits hardest. A 4 with no dots around it can't have a 2, 3, 5 or 8 beside it. A 1 or 9 only rules out one digit. When a cell has several blank borders, test the digits with the longest lists first. A 5 with blank borders only rules out 4 and 6, since 5 has no black-dot partner, and a 9 only rules out 8.
When 1 and 2 share a border
1 and 2 are both consecutive and in a 1:2 ratio, so they fit either color. Setters usually show one dot for this pair, but conventions vary. Read the rules to see which dot the puzzle uses.
Because of this, a white dot with a 1 on one side always has a 2 on the other, and so does a black dot. The 1 is the only digit whose partner is the same on both colors.
Common mistakes
Reading a black dot as a difference of 2
A black dot means one digit is double the other. 3 and 5 are 2 apart but fail a black dot. 3 and 6 pass, since 6 is double 3.
Using blank borders when not all dots are given
If the rules say not all dots are given, or say nothing about it, a blank border means nothing. Only use the negative rule when the puzzle states it.
Forgetting that 2 and 4 have two black partners
2 pairs with 1 or 4, and 4 pairs with 2 or 8. Every other digit on a black dot has only one partner. Solvers often assume a 4 pairs only with 8, or a 2 only with 4.
In mixed-rule puzzles
Kropki dots mix with many rules. A white dot pair and a V pair overlap only at 2 and 3, since 2 + 3 = 5. No pair fits both a white dot and an X, since two consecutive digits add to an odd number. No black-dot pair adds to 5 or 10, so a black dot never shares a border with an X or a V. German whispers lines reject every dot pair, since the largest dot gap is 4 (between 4 and 8). A parity line only allows the black-dot pairs 1 and 2 or 3 and 6. Along a thermo, a white dot means the thermo steps up by exactly 1.
What the video shows
- A white dot joins consecutive digits. A black dot joins digits where one is double the other.
- 4 and 5 are one apart, so they fit the white dot.
- 3 and 7 don’t fit the black dot: 7 isn’t double 3.
- 3 and 6 do: 6 is double 3.
Questions solvers ask
What does a black dot mean in sudoku?
A black dot between two cells means one digit is exactly double the other, like 2 and 4 or 3 and 6.
What does a white dot mean in sudoku?
A white dot means the two digits are consecutive, like 6 and 7.
Can 1 and 2 go on either dot in Kropki sudoku?
Yes. 1 and 2 are one apart and 2 is double 1, so the pair fits a white dot or a black dot.
What does "not all dots are given" mean?
It means some consecutive or 1:2 pairs have no dot. You can't use a blank border to rule anything out.
What can go next to a 4 on a black dot?
2 or 8. 4 is double 2 and half of 8. The only other digit with two black-dot partners is 2, which pairs with 1 or 4.
Can 5 go on a white dot?
Yes. 5 can sit on a white dot next to a 4 or a 6. It can never sit on a black dot.
Is Kropki the same as consecutive sudoku?
Not quite. Consecutive sudoku usually uses only the white marks for digits one apart. Kropki adds black dots for pairs where one digit is double the other.
Stitched Rainbow
Record Highs
Regional Constraints
Stack 'em Up, Knock 'em Down
'leven
Caged by a Secret Killer