The contradiction move

For most easy and many medium nonograms you can finish without ever looking at a row and a column at the same time. The line-solving move pays out cell after cell, the runs converge, and the picture emerges. Then one day you hit a puzzle where every line is exhausted and the grid is still half empty.

This is where the contradiction move comes in.

The idea: pick a cell whose state you don't know, hypothesise it's filled, and trace what that forces along its row and its column. If filling the cell creates a contradiction in either line — a run that overlaps the wrong cell, or a row whose clue can't be satisfied with the new constraint — then the cell must be empty. The same logic runs the other way: if hypothesising empty creates a contradiction, the cell must be filled.

The technique sounds expensive and in practice isn't. You don't trace all the way to a finished puzzle; you trace one or two steps and look for a clue that won't add up. A run of 4 in the column that now has to start at a cell that puts its tail past the grid edge. A 2 3 row whose first run can no longer be placed because of the new filled cell. The contradictions surface fast.

There's a useful shorthand for spotting candidates. After every line-solving pass, look for cells that have neighbours in three of four directions decided. A cell whose top, bottom, and left are all settled is well-constrained — the column above and below tells you what the remaining vertical run can do, the row to the left tells you what it can do, and the contradiction usually pops out within a step. Cells with only one or two settled neighbours are worse candidates because hypothesising either state leads further before you get a contradiction or a confirmation, which means more time spent in your head.

The only real failure mode of the contradiction move is going too deep. If you've traced three steps and nothing has contradicted, abandon the line. The cell wasn't a candidate; pick a different one. Spending five minutes mentally simulating a chain of placements is how a hint button starts to look attractive, and once a hint button looks attractive the puzzle has won. Better to give up on one cell, return to the line-solving sweep with the new information you have, and let another candidate surface naturally.

In a well-tuned hard puzzle there's usually exactly one contradiction-move opportunity per stuck point. Find it; use it; the line-solving sweep that follows will pay out half a dozen more cells. The rhythm of a hard nonogram is not contradiction after contradiction but the contradiction that unsticks the line-solving, again and again, until the grid is full.