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Technique 8 of 8

The Contradiction Test: Proving Cells by What-If

Tentatively assume a cell is filled (or empty), follow only the forced consequences, and watch for an impossibility. If the assumption breaks the line, the opposite state is proven.

Sometimes no direct technique fires, yet the puzzle is far from ambiguous. The contradiction test — what-if analysis — converts a stall into a proof: tentatively assume a cell is filled, propagate only the forced consequences through its row and column, and watch for an impossible state. A clue with no room left, two runs forced to merge, a count that overshoots the line. If the assumption leads to impossibility, the cell is provably empty. Assume empty and break the line instead, and it is provably filled.

Here is a complete one-line example. Width 6, clues 2 2, and the third cell is already known filled. Is the first cell filled? Suppose it is: then the first run of 2 occupies cells 1-2, and a run of exactly 2 must be followed by an empty cell — but cell 3 is filled. The runs would merge into a block of at least 3, which no clue allows. Contradiction; cell 1 is empty:

2 2known so far
2 2assume cell 1 filled
2 2××contradiction: cell 1 is X
If cell 1 were filled, the first run (cells 1-2) would butt against the filled cell 3 and merge into a run of 3+. Marking cell 1 forces the first run to cells 2-3, and the rest of the line follows.

Notice what happened after the proof: with cell 1 marked, the first run must cover the known cell as 2-3, punctuation seals cell 4, and the second run has exactly cells 5-6 left. One contradiction unlocked the entire line. That payoff pattern is typical — use the test on cells whose state would ripple, usually where a row and column are both nearly resolved.

Two rules keep the technique honest. First, propagate only forced moves while testing; if you find yourself choosing between options inside the what-if, back out — that is guessing wearing a disguise. Second, keep the chain short. On this site every puzzle is generated and verified to have a unique, logic-reachable solution, so a one- or two-step look-ahead is always enough; if a deep chain seems necessary, a simpler deduction is sitting unnoticed in a crossing line. The contradiction test is the last tool in the box precisely because the others are cheaper — but when you need it, it is a proof, not a gamble.

Try it on a real board.