Heyawake

Paint cells by the room numbers — no two painted cells touching, white in one piece, and no white corridor running through three rooms.

What it is

A grid divided into rectangular rooms. Some rooms carry a number: exactly that many cells of the room must be painted. Painted cells never share an edge; the unpainted (white) cells stay connected; and — the rule the puzzle is named for — no straight run of white cells may pass through three rooms.

How to play

Numbered rooms are arithmetic: a 2×2 room clued “2” paints a diagonal. The corridor rule is the real game — every straight line of white must be broken before it crosses its second room border, which forces paint in rooms with no number at all. Alternate the two: paint from numbers, then scan rows and columns for corridors about to run too far.

Purpose

The workspace’s demonstration that rooms can be the clue: a typical board ships with fewer than half its rooms numbered, and the geometry carries the rest. It also produced a foundation-level win — profiling its search exposed a shared dilate helper costing 110 µs per call, whose four-shift rewrite sped up every shading puzzle in the workspace.

History

Nikoli, 1992, by Hiroyuki Fukushima. Heya wake means “divided rooms”. Its corridor rule makes it one of the most theory-rich Nikoli puzzles — published boards lean on parity arguments researchers still write papers about.

This implementation

• Spec knobs: rows, cols, max_room, difficulty.
• Generation: rooms by guillotine cuts (rectangles by construction), an answer found by the same backtracking search that checks boards, then densified — cells painted until no more fit — because sparse answers are not pinned by their counts (a fully numbered sparse board still had five answers). Numbers are read off the answer and thinned while unique.
• Guarantees: exactly one painting satisfies the numbers, and it is the painting they were counted from. The room-by-room search is cross-checked against brute force over all 65,536 paintings of a 4×4.
• Measured, not guessed: the room cap decides feasibility — at 8×8, cap 4 left 12/24 boards uniquely determined, cap 6 left 2, cap 9 left 1. Small rooms constrain more. Difficulty is the share of rooms still numbered (rating_basis: numbered_room_share), thresholds set from measured quintiles.