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Neighbours

A group of people has decided to buy a piece of land. They want to divide the land equally, and each live on their own land. However, some people are more social than others: everyone has a preferred number of neighbours in the new situation. Furthermore, everybody has already claimed one square rod of land.

Divide the field into a number of connected regions of an equal number of unit squares, such that every region contains exactly one square with a number (or a question mark) in it, and has as many neighbours as the number indicates. Fields are neighbours when they share a part of their border.

Example

In the example we need to make four regions. Two of them need to have three neighbours, so they must be connected to all other regions. This means that the two twos must be connected to the two threes, and therefore not to each other.

Since the twos are not connected, they must be separated by one of the threes. The only way in which this is possible without locking the bottom right two in, is by extending the top three one square to the right and to to the top.

The region of the top three now consists of four squares, so it is finished. The four squares on the right can only belong to the two in the bottom right, so they also form a region together. Now the remaining eight squares can only be divided into two regions in one way.


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