Nurikabe

Read the numbers as islands, the rest as one sea

Nurikabe asks you to split the grid into two things: little islands of unshaded cells, each labelled with its exact size, and a single connected shaded sea flooding everything else. Four rules do all the work, and holding them in mind at once is the real challenge. Each numbered cell anchors an island of precisely that many cells. Islands never touch each other edge to edge. The sea is all one piece; you could walk it end to end without stepping onto an island. And the sea may never form a solid two-by-two block, the pool that gives the puzzle its bite. Every well-made Nurikabe has exactly one shading that satisfies all four at once, so every cell is provable. The skill is learning which proofs come cheap.

The free moves every solver plays first

A handful of openings cost nothing and appear in nearly every puzzle. Play them on sight.

• A clue of 1 is finished the moment you see it. Its island is just itself, so all its neighbours are sea.
• Any cell wedged between two different clues must be sea, because if it belonged to either island the two would touch. This fires for diagonal clues too: when two clues sit corner to corner, the two cells they both neighbour must each be sea, since joining either island would make the islands touch.
• Cells that no island could reach without overshooting its size must be sea. A clue of 2, for example, can only ever occupy its own cell and one direct neighbour, so anything two steps away is beyond it.
• When a partly built island has only one direction left to grow, it grows there. No choice means no guess.

One more workhorse deserves a name: the overlap. When an island has a few possible shapes, sketch them all in your head and look for cells that appear in every single version. Those cells are island for certain, even though you don't yet know the final shape. A clue of 3 sitting one cell out from a corner is the classic case: however its three cells bend, some cell is shared by all the options, and you can mark it immediately. The same idea works in reverse for the sea, since a cell in none of the possible shapes must be shaded.

Let the sea's own rules finish the job

Mid-puzzle, the two sea constraints start doing the heavy lifting, pushing cells from both sides. Watch for the pool squeeze: three shaded cells forming an L around a two-by-two square force the fourth cell to be island, because shading it would complete a forbidden pool. Then watch for the strangle: if shading a cell would cut the sea into two pieces, that cell must be island, and if leaving a cell unshaded would strand a pocket of sea from the rest, it must be shaded. Here's how the pressures combine in practice. Say a 2-island has been placed and sealed, and beside its ring of sea a lone unshaded cell sits with sea on three sides. If no clue can reach that cell, it must be shaded, but if shading it would build a pool, something upstream is wrong and one of your earlier cells is misplaced. Contradictions like that are the puzzle talking to you. A finished island seals itself with a ring of sea, and each sealed island simplifies everything around it, which is why the endgame usually collapses in a rush after a slow middle.

The wall that blocks the road

Nurikabe takes its name from a creature of Japanese folklore: an invisible wall-spirit said to stretch across a traveller's road at night and refuse to let them pass. Walking around it is hopeless, since the wall extends as far as you do, but folklore offers a fix: strike the base of the wall with a stick and it vanishes. The manga artist Shigeru Mizuki drew the creature as a sleepy slab of plaster with little limbs, and that image made the nurikabe a minor celebrity in Japan. The puzzle borrowed the name because solving it is literally painting a wall; it comes from Nikoli, the Japanese publisher behind many of the best logic grids, where it first ran in the early 1990s. Fans sometimes call the puzzle Islands in the Stream, which describes the finished picture perfectly: white islands, sized to order, floating in one continuous strip of ink.