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Stream Islands

A group of islands is located in a heavily branching stream. Can you find out where the islands are?

Each cell in the diagram is either part of an island, or water. Islands are connected regions surrounded by water, and all the water connects to form the stream. No lakes are allowed: in any 2x2 larger square, there must be at least one land square. Each island contains exactly one cell with a number in it, and this number indicates the size (number of cells) of the island.

Example

The island of size 3 is very much restricted by the other islands: From the square with the 3 in it, it does not extend either south or west, because that would connect it to another island. Also it can not extend to the north because in that case there is no way to complete the island of size 2 in the upper right corner. It follows it extends to the east. From there the only possibility is to add a square to the south. Consequently the island in the upper right corner extends westwards.

Focussing on the 2x2 square in the lower left corner, we notice there is just one way to avoid a 2x2 lake. The island of size 2 extends to the west. For a similar reason the remaining island extends to the south. All remaining squares are water that is connected, so this solution is valid.

Background

"Stream Islands" or "Islands in a Stream" was originally invented in Japan, and is also known under the name "Nurikabe" (ぬりかべ): a Japanese spirit that manifests itself as an ever-extending invisible wall.


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