How to solve Binary Signs

Put a filled or a hollow square in every empty cell so that each row and each column holds as many of one as of the other.

Never three alike side by side — not in a row, not in a column.

An = between two cells says they hold the same; an × says they do not.

Tap a cell for a filled square, tap again for a hollow one, once more to clear it. The squares the board came with stay put: tap one and they all give a shake, to show which they are.

Shown in red: three of the same kind side by side, squares a row or column has too many of, and two cells that disagree with the sign between them.

A few squares are given at the start and those stay put; the rest of the cells are empty, and every one of them is yours to fill. Signs stand between some cells only; where there is no sign, two neighbours may be alike or may differ. Two rows or two columns may come out identical — the rules do not forbid it.

Every board has exactly one answer, and it can be reached without guessing — when nothing stands out, there is always some sign, row or column with only one way left to go.

Two of the same symbol side by side is the most common shape on the board. A third of the same cannot stand on either side of them — so both neighbouring cells get the other symbol. It works the same way in rows and in columns.

Two of the same symbol with a single empty cell between them are a pair that does not know it yet. A third of the same in the gap would make a run of three — so the gap gets the other symbol.

On a 6 × 6 board every row and every column has exactly three filled squares and three hollow ones. A line that already has all its hollow squares has no choice: every empty cell in it is filled. Pairs and gaps usually leave exactly such tails behind them.

A sign with something already standing on one side of it is a ready answer: an = puts the same symbol on the other side, an × the other one. Signs like these are usually where a board begins.

A sign says something even while both of its cells are still empty. Two cells joined by an = are a pair before anything stands in them — and a pair cannot sit next to a third of the same symbol. Two cells joined by an × are always one filled square and one hollow, whichever is which — and that can already be counted.

When pairs, gaps, counting and signs have nothing to say, what is left is to read a whole line: how many filled squares and how many hollow it still lacks, and the ways they can be spread over its empty cells with no three alike together and every sign in the line kept. A cell that comes out the same in every one of those ways is settled — though no single move pointed at it.