Contradictions: try a square, and let the grid refute it

How a try works

Choose an unknown square and assume it is a box. Run every technique you know, on every line, as if the assumption were true. If some line ends up with no legal arrangement, the assumption was impossible, so the square is empty. If nothing breaks, you learn nothing and must rub the pencil out, because the assumption is not proven either.

The reasoning is airtight only when you follow it to a break. A guess that merely looks fine proves nothing, and treating it as true is how errors spread across a grid.

Choosing a square

Most squares lead nowhere. Good candidates are squares next to many known squares, squares near a border or a block of crosses, and squares in lines that are already mostly filled. Those have the fewest ways out, so a break arrives sooner.

Use a different colour for the assumed squares so you can undo them cleanly. In the practice grid below the solver proves a square this way, and the page shows which line broke.

The move in three steps

1. Choose a square. Pick an unknown square with many known neighbours.
2. Assume and follow. Pencil in a value and run every technique on every line.
3. Look for a break. If a line cannot be satisfied, the opposite value is certain. If not, undo it all.

Worked example: a try that fails

Row 2 has the clue 2 and a box on column 5. Suppose you pencil a box in column 1 of that row. The row now holds boxes in columns 1 and 5, four columns apart, but its only run is two long. That is impossible, so the pencilled box is wrong and column 1 is empty.

Had the pencilled square been column 4, nothing would have broken, and you would have learned nothing from the try.

Common slip. Keeping a guess that did not break anything. No contradiction means no proof, so the pencil marks must go.