Yajisan-Kazusan

Shade cells so that no two shaded cells touch and the unshaded cells stay in one connected piece. Each clue counts the shaded cells its arrow points at — but only if its own cell is unshaded. Shade the clue's cell and the number is excused from being true. Click a cell to cycle shaded → unshaded → blank; right-click cycles the other way.

shaded — its clue is excused pink number — no arrow on that line could ever show it

The clue is not a statement about the grid

In almost every arrow puzzle, a clue is an assertion: this many shaded cells that way. Here the assertion is conditional on one of the unknowns — the clue's own cell. If that cell ends up shaded, the number is never checked, and it may be anything at all. So a Yajisan-Kazusan clue is a disjunction: either I am shaded, or my number is right.

The useful half of that is the contrapositive. If the number cannot possibly be right, the cell must be shaded. A line of L cells can show at most ⌈L/2⌉ shaded ones, because they may not touch, so any bigger number is a confession. On the boards shipped here, adjacency on its own decides 0% of the grid — there is nothing shaded yet for it to work from — and the very next rung, which does nothing but catch clues lying, jumps straight to about a third. Every single thing this solver learns first, it learns from a liar.

A clue set of nothing but liars — and exactly when it works

Write an impossible number on every shaded cell of an answer. Each one confesses, so all of those cells are forced shaded; nothing else is said. Any other answer must therefore shade at least the same cells. That pins the answer down precisely when no cell can be added to it, and it provably fails when one can — because every clue would sit on a cell the bigger answer also shades, where every clue is excused. So: an all-liar clue set exists if and only if the answer is maximal. The repository checks both directions on 540 answers, and on the maximal ones the all-liar set is not even more expensive than a free choice of clues.

Five rungs

adj is the no-touching rule. cap is the confession. ray is the arithmetic once a clue cell is known unshaded, run as a small dynamic program over the line that also respects the no-touching rule inside it. white is the topology of the unshaded region, including cut cells. probe is singleton consistency on top — and on these boards it is startlingly strong: over 718 clue prefixes, "probe finishes without a guess" and "the board has exactly one answer" never once disagreed.