No more than two of the same colour next to each other, in a row or column.
Each row and each column must end up with the same number of black and white circles.
No two rows may match, and no two columns may match.
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How the Clue button reasons
The Clue button looks for these deductions, in order from simplest to most advanced:
Avoiding three in a row. If two neighbouring cells already share a colour, the next cell in line can't match, or it'd make three in a row.
Completing a row or column by count. Once a row or column already has its full share of one colour, every empty cell left in it must be the other colour.
Avoiding a duplicate line. If a row or column has one cell left, and filling it a certain way would exactly match another completed row or column, it has to go the other way instead.
Testing both colours. For a tougher cell, the Clue button tries both colours and follows the logic forward — if one choice eventually breaks a rule anywhere on the board, the other one is forced.