These are three-dimensional Nurikabe puzzles. In each case, the four squares represent the layers of a $4\times4\times4$ cube. The goal is to shade some cells in each layer so that the resulting space satisfies the rules1 of Nurikabe:
- Numbered cells cannot be shaded.
- Unshaded cells are divided into regions, all of which contain exactly one number. The number indicates how many cells there are in that unshaded region.
- Regions of unshaded cells cannot be adjacent to one another, but they may touch at a corner or along an edge.
- Shaded cells must all be orthogonally connected in 3D space.
- There are no groups of shaded cells that form a $2\times2\times1$ cuboid in any dimension.
1 Paraphrased from the original rules on Nikoli
In all three explanations, I'll use "LxRyCz" to refer to layer X, row Y, column Z (all counted starting from 1, left-to-right or top-to-bottom). The directions will be "left/right/up/down" within a layer, and "back/forward" between layers -- the first layer is the "front", and the last layer is the "back".
The obvious place to start is with the size-1 regions:
Next, some empty cells can only be accessed by certain regions:
And we've completed more regions, and forced some more unshaded cells:
and the rest resolves with the same techniques.
Start with the same techniques as before: finish the size-1 regions, and mark walls in any cells that would connect two regions.
Some regions now have only one way to extend:
Now we've incidentally finished a region by forced empty cells: block it off and mark any newly arising almost-2×2×1s.
Finally, there's one last deduction to finish the puzzle off:
Something has to reach the bottom-left of the front layer. The only region that can do that is the 5, meaning it has to go up through L(2-3)R4C2. That takes up four of its five cells:
and the remaining one has to be used to block a 2x2x1 in the back-bottom-right.
Once again, finish off the 1-regions and shade any cell that would connect two rooms.
Some regions have only one way to extend now: do that.
Now, there's a...
cell that can't be reached: the very front-top-left cell. Once that is taken care of, and the front 3 extends upwards into L1R2C2, there are a few more unreachable cells in layer 1, and one in layer 2 at L2R2C3.
The last cell in the previous step forms another almost-2×2×1. This completes a region:
The rest of the puzzle resolves just by repeatedly completing regions and checking for new unshaded cells.