Nurikabe

Nurikabe Example

Nurikabe is a puzzle type made by nikoli.

Cell types

In an unsolved puzzle, there seems to be only one type of cell, white cells, though some white cells have numbers in them. However, in the solved puzzle, a new type of cell appears: black cells. White cells still exist in the solution; in fact, all of the cells with numbers are white.

Interaction

The black cells divide the grid into separate parts of white cells. Each of these parts is called an “island”.

An island must have exactly one number, and its area is equal to the number. And no island may border another island. They may touch at corners, though.

The black cells also never form a group of 2×2 black squares, and all black cells are connected, not counting diagonally adjacent as connected. So if R3C3 was white, there will be two groups of black cells which is forbidden.

Objective

Shade some cells to be black such that the above conditions are satisfied.

Walkthrough to the example

Begin with R2C1. A 1 is always “full”; all the cells bordering it must be black. So R1C1, R2C2, and R3C1 are all black. Moreover, since black cells must be connected, R1C2 must be black.

Note that R1C3 is unreachable by any number, and thus must be black.

Now, if R2C3 is black, then there’s a group of 2×2 black cells, violating the rules. So R2C3 is white. The only possibility is for the 3 to extend maximally to R2C3, so R4C4 and R1C4 must be black as it’s now no longer reachable.

Since R4C4 is black, it must be connected to other black cells, so R4C3 and R3C3 are black. This makes the 3 to extend upward, so R2C4 is black.

Last, the 2. If the 2 extends upward, the black cell on R3C1 is isolated along with R4C1. So the 2 must extend leftward, making the solution.