Deeper search: when one proof is not enough

Stacking proofs

In a contradiction you assume one square and watch the lines. On harder grids that single proof pays out only a little: the lines move, then stall again. So you pencil the next proof, and this one is allowed to use everything the first one taught you. The practice grid below is built so that line logic alone stops almost at once, and a solver has to chain several proofs one after another to finish it.

The discipline is to finish each proof before you start the next. Write its result in ink, run every line rule again, and only then look for the next square to test. Mixing two unfinished guesses is how a grid gets ruined.

A level down: a try inside a try

The next step is a try inside a try. You assume one square and run the line rules, and nothing breaks. Inside that world you then assume a second square. If that second assumption breaks a line, the second square is refuted, which gives you a new fact inside the first world. A few of those facts together can break a line, and then the first assumption is refuted.

The cost grows fast, and Wikipedia says it is not possible simply by pen and pencil. We went looking for a small grid that needs it. We checked 500,000 distinct 6x6 and 7x7 grids with a solver, including ones picked for being close to ambiguous. None of the grids with exactly one solution needed more than one level of trying. That is a sample of the possible grids, not a proof that none exists.

The move in three steps

1. Prove one square. Use a contradiction to settle one square, then finish all the line logic it unlocks.
2. Repeat. When the lines stall again, pencil another proof. Each proof may use every square the earlier ones settled.
3. Go a level down last. Only if no single try breaks anything, assume a square and try a second one inside it. Expect that in large hand-made grids, not small ones.

Worked example: three proofs in a row

Line logic stops with only a handful of squares known. You test a square, and it breaks a row, so you write the opposite in ink and run the lines again. They move a little, then stall. A second test, which only works because of the first, settles another square. A third finishes the grid. Each proof is small. What makes it a search is that you have to be ready to start another when the last one runs dry.

Now imagine a grid with 60 squares still unknown, where no single test breaks anything. A one-level search would have checked 120 tries. A try inside a try would check each of those and then up to 118 inner tries, about 14,000 checks, each with its own line logic. That is trivial for a program and unthinkable with a pencil.

Common slip. Treating the chain as one big guess. Each proof must be finished, and its result written in ink, before you start the next.