The Nonogram Contradiction Technique

When to use it

• Only after line logic has truly stalled. Re-check every row and column first, including the moves in Nonogram techniques beyond overlap; they are faster than any chain.
• On harder boards. In Nogramix, every Spring Forest puzzle and every Daily board rated Easy or Medium can be finished with line logic alone, so you will not need this technique there.

How it works, step by step

1. Pick an open cell. Good candidates sit in short or nearly finished lines, or decide where a block goes.
2. Suppose a value, filled or empty, and remember that it is only an assumption.
3. Follow forced deductions only. Apply ordinary line logic to the lines through that cell, then to the lines crossing anything that changed.
4. Look for a broken clue: a line whose known cells no longer fit any placement of its clue, such as too many filled cells or a block with no room.
5. If a clue breaks, undo every cell from the chain and write down the opposite of the supposition. That one cell is proven.
6. If nothing breaks, undo everything. The supposition might be right or wrong; you have learned nothing yet. Try the opposite value, or a different cell.

Inside a single line, it is just line logic

The idea works inside one row too: a cell is empty exactly when supposing it filled leaves no way to place the clue. Line logic already does this for you. Take a seven-cell row with clue 3 and its fourth cell filled:

So cell 1 is empty, exactly as line logic said. The technique earns its keep when the chain crosses into other lines, as in the next example.

Worked example: a stalled 5×5

This board has exactly one solution. It was made for this guide rather than taken from the game. Repeating line logic over every row and column stops here, with eight cells still open:

Test the top-left cell. The top row's clue is 2 and its second cell is filled, so the 2 sits either in cells 1–2 or in cells 2–3. Either value of the top-left cell has immediate consequences.

1. Suppose the top-left cell (row 1, column 1) is filled.
2. Row 1, clue 2: the block is cells 1–2, so cell 3 is empty.
3. Column 3, clue 4: row 1 is now empty, so the 4 must fill rows 2–5. The bottom cell is filled.
4. Row 5, clue 1 2: columns 3 and 4 are now filled side by side, so they are the 2. The 1 must come before it with a gap, which leaves only column 1. Column 5 is empty.
5. Column 1, clue 1: rows 1 and 5 are both filled. Two filled cells cannot be one block of 1, so the clue is broken.

The supposition is impossible, so the top-left cell is empty. Undo the whole chain and keep only that cell. Ordinary line logic now takes over: the top row's 2 must be cells 2–3, column 3's 4 fills rows 1–4 and leaves the bottom cell empty, and the remaining rows and columns resolve one by one.

At no point did you keep a cell you could not prove. That is the difference between a contradiction check and a guess, which Do Nonograms require guessing? looks at in more detail.

Choosing a good cell

• Cells in short or nearly finished lines. Fewer placements means the chain breaks, or fizzles out, quickly.
• Cells that decide where a block sits, like the top-left cell above, where the 2 had only two possible places.
• Cells next to filled cells. Supposing them filled often makes a block too long.
• Both values. If one supposition leads nowhere, try the other: proving the opposite is just as useful.

Common mistakes

• Keeping cells from a failed chain. Only the opposite of the supposition is proven; everything else in the chain was built on a false premise.
• Treating “nothing broke” as proof. If no clue breaks, the supposed value may still be wrong.
• Skipping line logic. A chain is slower than reading a line. Use it only when every row and column is genuinely stuck.
• Following hunches inside the chain. Each step must be forced by a clue, or the result proves nothing.

Play today's Daily Nonogram