# KROPKI SUDOKU

Kropki is Polish and means “little dot”. In this variant, in addition to predetermined numbers, there are also dots between the cells that determine their relationship to each other.

So a filled dot means that the neighboring cell has either double or half the value and an unfilled dot stands for one more or fewer. None of these rules apply to all other cells.

# Terms

## Filled point

A filled dot between two cells indicates that the neighboring cell has double or half the value.

Example: the neighboring cell of a 4 can only be a 2 or 8.

## Filled point

A filled dot between two cells indicates that the neighboring cell has double or half the value.

Example: the neighboring cell of a 4 can only be a 2 or 8.

## Circle

Two cells with a circle (open dot) between them are always one larger or smaller.

Example: the neighboring cell of a 3 can only be a 4 or 2.

## Circle

Two cells with a circle (open dot) between them are always one larger or smaller.

Example: the neighboring cell of a 3 can only be a 4 or 2.

## Without marking

If there is no marking between two neighboring cells, neither of the two rules mentioned apply.

Example: the neighboring cell of a 4 cannot be 2, 3, 5 or 8.

## Without marking

If there is no marking between two neighboring cells, neither of the two rules mentioned apply.

Example: the neighboring cell of a 4 cannot be 2, 3, 5 or 8.

# Rules

In addition to the mentioned markings between the cells, the standard rules for Sudoku puzzles apply to Kropki. You can find these in the How to play Classic Sudoku. Before following the solution steps, it is recommended that you first familiarize yourself with these rules.

In addition to the mentioned markings between the cells, the standard rules for Sudoku puzzles apply to Kropki. You can find these in the How to play Classic Sudoku. Before following the solution steps, it is recommended that you first familiarize yourself with these rules.

# Solving Steps

## Step 1

In this example of a 4x4 Kropki Sudoku, the number 4 is given at C1 and the number 3 is given at C3. We also see that between the givens there is a solid dot between C1 and C2 and a circle between C2 and C3.

## Step 1

In this example of a 4x4 Kropki Sudoku, the number 4 is given at C1 and the number 3 is given at C3. We also see that between the givens there is a solid dot between C1 and C2 and a circle between C2 and C3.

## Step 2

This means we can enter 2 between these two numbers because it is half the size of 4 and therefore does not violate the rule for filled dots. In addition, the 2 is one smaller than the given 3 and therefore does not violate the rule for circles between cells.

## Step 2

This means we can enter 2 between these two numbers because it is half the size of 4 and therefore does not violate the rule for filled dots. In addition, the 2 is one smaller than the given 3 and therefore does not violate the rule for circles between cells.

## Step 3

With this we can complete this column with the missing 1.

## Step 3

With this we can complete this column with the missing 1.

## Step 4

The neighboring cell of the 1 we just entered at C4 is defined as double or half by the filled dot. For 1, only double the value 2 for B4 can be considered.

## Step 4

The neighboring cell of the 1 we just entered at C4 is defined as double or half by the filled dot. For 1, only double the value 2 for B4 can be considered.

## Step 5

The upper neighbor of the 2 at C4 is again double or half. Since a 3 is specified in cell C3, we can exclude the 4 for B3, otherwise we would violate the rule, that none of the rules may apply to unmarked cells. This leaves the only option for this cell to be 1 (as half the value of 2).

## Step 5

The upper neighbor of the 2 at C4 is again double or half. Since a 3 is specified in cell C3, we can exclude the 4 for B3, otherwise we would violate the rule, that none of the rules may apply to unmarked cells. This leaves the only option for this cell to be 1 (as half the value of 2).

## Step 6

For cell B2 as a neighbor of the 1 just entered at B3, we can exclude the 1. Since the two in C2 require a number that is twice or half as large, the only option remaining is 4. This means we can also complete the column with a 3 in B1.

## Step 6

For cell B2 as a neighbor of the 1 just entered at B3, we can exclude the 1. Since the two in C2 require a number that is twice or half as large, the only option remaining is 4. This means we can also complete the column with a 3 in B1.

## Step 7

Following the rules, we can enter a 2 at A1 (distance 1 to 3 in A2 and the 4 for this box already revealed). This is followed by a one at A2 and D1 to complete the respective line and box.

## Step 7

Following the rules, we can enter a 2 at A1 (distance 1 to 3 in A2 and the 4 for this box already revealed). This is followed by a one at A2 and D1 to complete the respective line and box.

## Step 8

The remaining numbers can be revealed by applying the rules of classic Sudoku and we can successfully solve the puzzle.

## Step 8

The remaining numbers can be revealed by applying the rules of classic Sudoku and we can successfully solve the puzzle. 