This is a Nurikabe puzzle. The goal is to paint some cells black so that the resulting grid satisfies the following rules.1
- Numbered cells are white.
- White cells are divided into regions, all of which contain exactly one number. The number indicates how many white cells there are in that region.
- Regions of white cells cannot be adjacent to one another, but they may touch at a corner.
- All black cells must be connected.
- There are no groups of black cells that form a 2 × 2 square anywhere in the grid.
1 Paraphrased from the original rules on Nikoli.
This seems to be the unique solution:
Here's my progress, step by step:
To start, surround the ones, prevent black 2x2s in the bottom corners (there's only one way to do that), and surround the lower twos:
Then, connect the black pieces that have only one way out, and mark a couple of squares that must remain white in order to prevent black blobs:
All the yellow-marked squares must be connected to one of the 20s. The one on the right cannot be connected to the lower one, because that would unavoidably split the chain of blacks. There's only one way to connect it to the other 20 without creating a separated black region, and that gives us more black squares, which again necessitates some white ones (which we mark yellow)
These define the remaining two and three, which have to be surrounded:
Then connecting the black bits with the minimum of black squares (without splitting the remaining 20-region) gives this result:
Which happily leaves exactly the required 20 white squares in the final region, so the solution is nice and unique.