Kropki Sudoku: Dot Rules and Solving Strategy

Read white and black dots correctly

Kropki Sudoku keeps the classic row, column, and 3 by 3 box rules. A white dot between two orthogonally adjacent cells means their digits are consecutive, so they differ by exactly 1. Possible white-dot pairs include 1 and 2, 4 and 5, or 8 and 9. A black dot means one digit is exactly double the other. Possible black-dot pairs include 1 and 2, 2 and 4, 3 and 6, or 4 and 8. The order is not specified by the dot, so either cell may hold the larger digit unless crossing logic decides it. Dots apply only to the two cells they touch. Diagonal cells are unrelated unless another rule connects them. Shapedoku itself follows classic shape Sudoku without Kropki dots, but its four difficulty levels are useful for practicing the accurate row, column, and box candidate work that this variant still requires.

Check whether missing dots carry information

Some Kropki puzzles use a negative constraint: if two adjacent cells have no dot, their digits are neither consecutive nor in a 1 to 2 ratio. Other puzzles show only selected dots, meaning an undotted pair may still satisfy one of those relationships. The puzzle instructions must state which convention applies. Never assume that no dot means no relationship unless the negative rule is explicit. This distinction can transform the candidate map. Under a full negative constraint, a placed 5 removes 4 and 6 from undotted neighbors, while a placed 4 also removes 2 and 8 because they form a doubling pair. Under a partial-dot puzzle, none of those undotted eliminations is valid. Read the rules before solving, because carrying the wrong convention through a grid creates deductions that look tidy and are completely fictional, which is a rather efficient way to ruin a quiet evening.

Turn dot chains into candidate ranges

Dots become especially powerful in chains. A sequence of white dots must rise and fall through consecutive values, while black-dot chains multiply or divide by 2 at each step. Since digits stay between 1 and 9, long black chains have very few possibilities. A three-cell black chain might use 1, 2, 4 or 2, 4, 8 in one direction, with reversals depending on the geometry. Cross-check those sequences against repeated digits in shared units. For white chains, use end bounds: if three cells must form consecutive steps, a middle value near 1 or 9 may be impossible because it cannot support both neighbors. Once one cell is limited, update every connected dot before searching elsewhere. The strongest workflow alternates dot logic with classic Sudoku logic. A dot narrows two cells, a box restriction fixes one, and that placement then collapses the next dot relationship.

1. Mark every white dot as a difference-of-one relationship.
2. Mark every black dot as a doubling relationship.
3. Confirm whether undotted neighbors carry a negative constraint.
4. List the legal digit pairs for each dot before choosing positions.
5. Propagate candidate changes through connected dot chains.
6. Reapply row, column, and box logic after every dot-based elimination.