Enter digits from 1 through n in every row and in every column, such that no row or column contains any digit more than once. Anywhere there are two horizontal or vertical neighbors, one of which is double the other, there is a black dot between them. Anywhere there are two horizontal or vertical neighbors that differ by one, there is a white dot between them. Between 1 and 2 there can be either a black or a white dot. Next to the puzzle the value of n is given.

On the top row the three leftmost squares can only be 1,2,4 or 4,2,1. This means the second square contains a 2 for sure and the fourth square contains a 3. Below the 2 there has to be a 3 or a 1. In case that would be a 1, below it there would have to be yet another 2, but that is not allowed. So the second column should read 2,3,4,1.

On the left of the 4 in the second column there must be a 2, while on the right there must be a 3. So the third row should read 2,4,3,1. Above this 1 there must be a 2 and below it can only be a 4. In this fashion the whole grid can be completed.

Kropki is Polish for "dot" or "point".

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