Nikoli-style pencil puzzles

Pencil puzzles in the Nikoli tradition are text-free: the grid and a short geometric or topological rule are enough. You do not need Japanese to play; you need the invariant (loop, shade, bridges, rooms).

Connect adjacent dots into one simple loop. A number in a cell says how many of that cell’s four edges are on the loop. No branching, no crossings.

Shade a connected “sea” so numbered cells seed white islands of the given size. The sea usually cannot contain a 2×2 block; islands do not touch orthogonally.

Link numbered islands with orthogonal bridges. Bridge count matches the number; bridges do not cross; at most two bridges between the same pair on typical rulesets.

Draw one loop through cells. White pearls: the loop goes straight through and turns in the cells before/after. Black pearls: the loop turns on the pearl and goes straight in the neighbors.

Draw one loop and place black blockers. Blockers do not touch orthogonally; arrow clues count blockers in that direction. The loop visits every non-blocker white cell under common rules.

Paint black cells so room clues are satisfied, white cells stay connected, and a straight white run does not cross too many room borders in one line (classic Heyawake bound). Blacks do not share an edge.

A single closed loop means every used vertex has degree two; a dead-end edge can be marked off early. Parity across a cut (odd unfinished exits) often proves a branch impossible — useful whether you solve by hand or with a connectivity tool such as Noqx.