Trial, and Why We Throw It Out

It cracks every board ever made, which is the problem with it. Here is how to do it properly if you meet one elsewhere, and why nothing here will make you.

How a trial actually works

Choose an undecided link and provisionally give it a number of bridges. Follow the consequences with every ordinary technique — forced islands, closures, crossings, capacity, connectivity — until either the board finishes or something breaks: an island with more bridge ends than its number, two bridges crossing, a group sealed off with islands outside it. If it breaks, that number was wrong, and ruling it out is a sound permanent deduction.

The discipline is all in the bookkeeping. Everything derived under a trial is provisional and has to come back off if the trial fails, so it has to be visibly distinguishable from what you knew before — a second colour on paper, or the undo button here, which is precisely what it is for. A solver who cannot cleanly unwind a trial will eventually leave a consequence of something false on the board, and that is a far worse position to be in than being stuck.

Choose the link carefully if you are going to do it at all. The best candidate touches an island with almost no slack, because the consequences arrive within a move or two and the branch resolves quickly. Trying a link out in a sparse corner produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies

After — one link, proved by contradiction

This board has exactly one answer and no reasoned route to it — the kind this site grows, grades and then discards. Seven ordinary moves have been made and nothing above is available: no island is forced, none is finished with links open, no crossing is left, the arithmetic is exhausted and no group is close to sealing. Supposing the top-row link empty breaks the board, so it carries a bridge. Sound, and thoroughly unsatisfying.

what it decides what it wrote

Why it cannot be a difficulty level

Because it works on everything without exception. Any board with a unique answer can be finished by trying, following and unwinding, repeatedly if necessary, so "needs a trial" is a statement about the solver’s patience rather than about the board. A scale whose top rung fits every board is not measuring anything.

It is also what a generator falls back on when it has stopped checking. Grow a network, read the numbers off it, verify only that the answer is unique, and you get boards that are unique and unreachable by reasoning — which is the defect behind almost every "impossible" review a bridges app collects. The board is not impossible. It is simply not a puzzle any more, and the app cannot tell, because it never asked.

So the generator here asks. Every board is put through the technique solver as well as the counter, and any that needed a trial is thrown away and another grown in its place. The position below is what one of those rejected boards looks like part-way through: seven ordinary moves have been made, everything ordinary has run out, and only a trial will move it.

If a board elsewhere forces one on you

Try into the most constrained link you can find, keep the provisional bridges visibly separate, and stop the instant you hit a contradiction rather than pressing on to see how bad it gets. One resolved link is a complete result — go straight back to the cheap techniques, because that link will usually have closed something out and forced an island or two.

And if you find yourself running a trial inside a trial, put the board down. Two levels of provisional working is where the errors come from, and a board that requires it was produced by something that never checked whether a person could follow it.

Where this sits on the ladder

Rung 1Full house
An island with one open link, or with only just enough room, has no choice at all.
Rung 2Spent island
An island that already has every bridge it asked for closes all its other links.
Rung 3Blocked crossing
A link that would cut across a bridge already built can never be used.
Rung 4Spare capacity
Take one link away and ask what the rest could carry; the shortfall is forced down it.
Rung 5Isolation
A group cannot seal itself off, and a group with one way out has to use it.
Rung 6Assumption
Draw a bridge, follow it until something breaks, and rub it out again.

Common questions

Is trial and error allowed in a bridges puzzle?

It is sound reasoning — a supposition that leads to a broken rule proves the opposite. But it works on every board, so a board that requires it has not been checked for human solvability rather than being especially hard.

Do any boards on this site need a trial?

No. Every board is graded by the technique solver before it is published, and any that reached the trial rung was discarded and replaced rather than sold as a harder level.

Why do some bridges apps produce boards that feel impossible?

Because they check only that the answer is unique. Unique and humanly reachable are different properties, and a generator that tests the first without the second will ship boards that can only be finished by trial.