Where a Cage Meets a Line

Every rung below looks at one cage or one line. This one holds both at once, and it is the last argument any board here is allowed to need.

A cage that traps a digit

Work out every arrangement of a cage, as usual, but this time look at which digits land in the part of the cage that sits inside one particular row. If some digit appears there in every single arrangement, then that row’s copy of the digit is inside the cage wherever it ends up — and every cell of the row outside the cage loses it.

The striking thing is how little has to be known for this to work. The cage may have four arrangements and not a single decided cell; nothing about the cage itself is settled by the argument at all. What it settles is somewhere else entirely, in cells that may belong to three other cages, and it does so on the strength of a fact about all four arrangements at once.

The smallest useful case is a two-cell cage lying flat inside a row. A 1− cage there holds two consecutive digits, and if its cells are down to three or four and the pair must be adjacent, the middle digit is in the cage in every arrangement. That digit then leaves the rest of the row, which regularly decides a cell three or four columns away.

Before — a 1− cage lying flat across the top row
	1	2	3	4
1	cage with a difference of 1234	234	cage adding to 713	12
2	cage adding to 121234	1234	13	12
3	12	12	cage adding to 114	3
4	13	cage with a difference of 113	2	4

After — the trapped digit leaves the rest of the row
	1	2	3	4
1	cage with a difference of 1234	234	cage adding to 71
2	cage adding to 121234	1234		12
3	12	12	cage adding to 114	3
4	13	cage with a difference of 113	2	4

The 1− cage holds two consecutive digits and its cells are down to two, three and four. Consecutive pairs from that set are two-three and three-four, so a 3 sits inside the cage whichever pair it turns out to be. The top row therefore has its 3 already spoken for, the cell three columns along loses it, and that cell had only one other candidate.

what the reasoning read what it decides what it wrote

Cells that share their candidates

The second half of the rung is the same idea with no cage involved. If two cells in a row hold only a 3 or a 7 between them, then whichever way round it goes, the 3 and the 7 are both spoken for — and every other cell in that row can lose both. Three cells holding three digits between them work identically, and three is about as far as anybody spots by eye.

It is worth being clear that this decides nothing directly. Nothing is written, nothing about the pair is resolved, and the board looks exactly as unfinished afterwards. What changes is everything else in the line, and on a board that has been stuck for five minutes that is usually where the next move was hiding.

These two belong on one rung because they are one habit. Both ask what a small group of cells must hold BETWEEN them rather than what any one of them holds, and both spend the answer on cells outside the group. A solver who has the habit finds them in either form; a solver who does not finds them in neither.

Before — a 14+ cage covering a whole column and one cell beside it
	1	2	3	4
1	cage adding to 141234	cage adding to 131234	cage multiplying to 412	14
2	1234	1234	1234	12
3	1234	1234	1234	cage adding to 81234
4	1234	1234	1234	1234

After — the cell outside the column is decided
	1	2	3	4
1	cage adding to 141234	cage adding to 13123	cage multiplying to 412	14
2	1234	123	1234	12
3	1234	123	1234	cage adding to 81234
4	123	4	123	123

This cage swallows the entire first column and one extra cell in the bottom row. The column holds a complete set of digits, which on a board of side four adds to ten, so the extra cell has to make up the difference between ten and fourteen. There are five open cells here and the cage decides one of them without anything else on the board being known at all.

Why nothing here climbs any higher

Above it there is only trial — write a digit, follow it until the board breaks, rub it out — and that will finish any board at all, which is exactly why it cannot be a difficulty level. A scale whose top rung fits everything measures nothing.

So expert here means precisely this argument and no more: somewhere on the board there is a cell that nothing simpler will decide, and what decides it is a cage and a line considered together. Boards needing anything beyond that are stopped during generation rather than sold as a harder level.

On a small board expert is genuinely hard to produce. Sixteen cells often do not leave a cage enough room to trap anything, which is why a 4×4 asked for at expert quite frequently comes back labelled hard. That is the label the solver measured, and it ships under that rather than under the one requested.

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

What does it mean for a cage to trap a digit?

Every arrangement of the cage places that digit somewhere inside the part of the cage lying in one row or column, so the line’s copy of the digit is inside the cage and every other cell in the line loses it.

Is a shared pair of candidates the same as a naked pair?

It is the same idea, yes. Two cells in a line holding only two digits between them lock both of those digits into the pair, and the rest of the line can be stripped of them.

Why does this technique never decide the cage itself?

Because it argues from what all the cage’s arrangements have in common, and if they had enough in common to settle the cage the cheaper cage-listing rung would already have done it.