# Kropki

## Rules

Place a number from 1 to X (integers only) into each cell so that each number appears at most once in each row and column. (X is the number of cells in each row.) A white dot on the edge of two cells indicates that those two cells must contain consecutive numbers; a black dot on the edge of two cells indicates that a number in one of those cells is double the value of the number in the other cell. If 1 and 2 are in adjacent cells, then the dot between them could be either colour. If there is no dot on the edge of two cells, it means neither a black nor a white dot could go there.

(Rules and example from PGP IB)

## History of the puzzle

Invented by Vladimir Portugalov (Belarus) in 2005. The name Kropki comes from Belarusian кропкі, meaning "dots."

First appeared in Forsmarts contest as an instructionless puzzle.[1] The name was probably given when this puzzle appeared on WPC 2006/Part 2.

## Variants

WPC 2016/Round 3 was a round solely consisted of Kropki variants.

### Kropkuro

See Kakuro#Kropkuro.

### Outside Kropki

Probably first appeared on Nikola Živanović's (Serbia) set from the 13th 24-Hour Puzzle Championship (2012).[2]

Fill in the whole grid with digits 1 to N (where N is the size of the grid) so that each row and column contains each digit exactly once. If there is a white dot between a pair of orthogonally adjacent cells, then the cells must contain numbers whose difference is exactly one. If there is a black dot between a pair of orthogonally adjacent cells, then the cells must contain numbers whose quotient is exactly two. There can be either a black or white dot between 1 and 2.

All dots from each row or column have been removed from the grid. They must appear in the corresponding row or column in the given order. There may be some gaps between the circles. All possible dots have been given.

(Rules and example from WPC 2016 IB, circles inside the solution grid are just for illustrative purposes.)