Trial, and Why We Stop Short of It

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

What a trial consists of

Pick an undecided cell and provisionally give it one of its candidates. Follow the consequences with every ordinary argument — printed cages, cage lists, singles, row totals, cage interaction — until either the board finishes or something breaks: a digit twice in a line, a cage that cannot reach its target, a cell with nothing left in it. If it breaks, that candidate was wrong, and removing it is a sound permanent result.

The discipline is entirely in the bookkeeping. Everything derived under a trial is provisional and has to come off again if the trial fails, so it must 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, which is a far worse place to be than stuck.

Choose the cell carefully if you are going to do it at all. The best candidate is a cell with two candidates inside a tight cage, because the consequences arrive within a move or two and the branch resolves quickly. Trying a cell with five candidates in a loose corner produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies anywhere
	1	2	3	4	5
1	cage with a difference of 1124	cage adding to 15124	cage adding to 524	cage multiplying to 1535	35
2	1235	1235	13	cage adding to 6123	single cell, 44
3	cage adding to 1212345	12345	12345	1234	123
4	12345	12345	cage adding to 1612345	12345	1235
5	12345	12345	12345	12345	1235

After — one candidate, removed by contradiction
	1	2	3	4	5
1	cage with a difference of 124	cage adding to 15124	cage adding to 524	cage multiplying to 1535	35
2	1235	1235	13	cage adding to 6123	single cell, 44
3	cage adding to 1212345	12345	12345	1234	123
4	12345	12345	cage adding to 1612345	12345	1235
5	12345	12345	12345	12345	1235

This board has exactly one answer and no reasoned route to it — the kind this site carves, measures and then throws away. Eight ordinary moves have been made and nothing above is available: no cage has a short enough list, no cell or digit has run out of room, the row totals say nothing new and no cage traps a digit. Supposing a 1 in the top-left corner breaks the board, so it is not a 1. Sound, and thoroughly unsatisfying.

what it decides what it wrote

Why trial cannot be a level

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

It is also exactly what a generator falls back on when it stops checking. Cut cages at random, verify only that the answer is unique, and out come boards that are unique and unreachable by reasoning — which is the defect behind almost every one-star review a cage puzzle app collects. The board is not impossible. It has simply stopped being a puzzle, and the app cannot tell, because it never asked.

So the generator here asks. Cages are merged one pair at a time and each merge is kept only while the technique solver can still follow the board without trial, which means the question is answered several hundred times per board rather than once at the end. The position below comes from a board carved deliberately past that point: eight ordinary moves have been made, everything ordinary has run out, and only a trial will move it.

If you meet one of these somewhere else

Try into the most constrained cell you can find, keep the provisional digits visibly separate, and stop the instant you hit a contradiction rather than pressing on to see how bad it gets. One eliminated candidate is a complete result — go straight back to the cheap arguments, because that elimination has usually shortened a cage list and decided a cell 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 requiring it was produced by something that never checked whether a person could follow it.

Where this sits on the ladder

Rung 1Free cells
A one-cell cage is its own answer, and a row with one gap left has no choice about it.
Rung 2Cage combinations
List every way a small cage can be filled and keep whatever all of them agree on.
Rung 3Singles
A cell with one digit left, or a digit with one cell left in its row or column.
Rung 4Unit totals
Every row adds up to the same known number, so the cages inside it price the rest.
Rung 5Cage interaction
What a group of cells can hold between them, whether they share a cage or only a row.
Rung 6Assumption
Write a digit, follow it until something breaks, and rub it out again.

Common questions

Is trial and error allowed in calcudoku?

It is sound reasoning — a supposition leading 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.

Does any calcudoku here need a trial?

No. The carve tests every cage merge against the technique solver and stops at the last merge the solver could still follow, so a board needing trial is never reached in the first place.

Why do some cage puzzle 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 testing the first without the second will happily ship boards that only trial can finish.