Light up every square: an unlit square holds a purple square and a lit one a yellow disc. Pressing a square flips a group of squares, each from one piece to the other. Which group depends on the game mode, and the Type menu has a section for each:
On a touch screen, a tap does what a click does.
Click a square to flip it and some of its neighbors.
With the keyboard, the arrow keys move a cursor around the board. Enter or Space does what a click does to the square under it.
In either mode, the diagram on each piece indicates which other squares will flip.
Pressing a square twice undoes it, and the order of presses never matters, so an answer is simply a set of squares to press once each. Hint takes the answer with the fewest presses and works through it left to right along the top row, then along the next row down. A square it has gone past is never pressed again.
That order is what gives a press a reason. Once the hint has gone past every other square that flips some unlit square, the one square left is the only one that can still light it, so it has to be pressed: “Row by row, only this square can still light the outlined square, so it must be pressed. It flips the striped ones too.” With Crosses this is the square directly below an unlit one, which is the whole method: get the top row right and each row after it is settled by the row above.
A striped square is not part of the reason. The press flips it along with the rest, and a square further on will set it right if it needs it.
The presses that come first, the top row’s with Crosses, have no reason of that kind: a square further on could still undo whatever they change. For those the hint says how many presses the board takes and that the ringed square is one of them: “The whole board can be lit in 5 presses, and no fewer. One of them: press this square.” When only one set of presses lights the board at all, it says so instead. And the last press says that it is the last: “Press this square: that lights the outlined squares and finishes the board.”
These parameters are available from the ‘Custom type…’ option on the ‘Type’ menu.