Working In From Both Ends

The second-cheapest thing on the board, and the one that turns a scattering of opening cells into a solve. It reads only how a line begins and how it ends.

A settled end makes a shorter line

Suppose the first two cells of a row have been crossed off. Then the row’s runs all have to fit into the eighteen cells that are left, and eighteen cells with the same numbers has less slack than twenty did. Recompute the overlap on what remains and cells that gave nothing a moment ago give something now. That is the whole idea, and it is why crossing cells off pays for itself immediately.

The same works from the other end and from both at once. Every cross at either end shortens the line; every filled cell at either end anchors a run to it. Only the settled stretch at each end counts — the moment your eye reaches an undecided cell, this rung stops looking.

That restriction is what makes it cheap. You are not analysing anything; you are noticing where the line really starts and where it really finishes, and redoing arithmetic you already know how to do.

Before — the second column is settled at its top
	1	21	21	2	1
21
2
2
11
1

After — the cell below it is filled
	1	21	21	2	1
21
2
2
11
1

The second column reads 2 1 and its first cell has already been crossed off, so its two runs have to fit into the four cells below. Four cells holding a two, a gap and a one is a perfect fit with nothing to spare, which settles the top of it at once. Only the settled start of the column was read; nothing further down was looked at.

what the reasoning read what it decides what it wrote

A filled cell at the very start

If the first cell of a line is filled, the first run begins exactly there. Fill along to the length of that first number and cross the cell immediately after it, because the run cannot be longer than its number says. It is the single most productive move in the puzzle after the opening sweep, and it is available on a great many lines the moment their first cell is decided either way.

The mirror image works at the far end with the last number, and the two are worth doing together — a line settled at both ends is often finished by this alone. On a line reading 4 2 whose first cell is filled and whose last cell is filled, everything except the middle is written down in two movements.

The trap is applying it to a filled cell that is near the start rather than at it. A filled cell in the second position, with the first still undecided, does not anchor anything, because the run might begin at either. Attaching a block that is not at an end is the argument two rungs above this one and it costs more for a reason.

Why this plus the opening is exactly what easy means

An easy board on this site is one the solver finishes with the numbers and the ends of the lines, and nothing wider. That is the definition rather than an approximation of it: the solver always takes the cheapest available step, and if it never needed to read the middle of a line, the board is easy and is labelled so.

It also means an easy board is guaranteed to be finishable by a reader who knows only this page and the one before it. If yours has stalled, the likeliest omission is a line whose numbers are all accounted for and whose remaining cells were never crossed off — every one of those is a settled end waiting to shorten a line somewhere else.

Where this sits on the ladder

Rung 1Overlap
A run longer than half the line has to cover the middle of it, wherever it sits.
Rung 2Working in from the ends
What the settled cells at the two ends of a line force, without looking between them.
Rung 3Gaps and splitting
The crosses cut the line into pockets, and each number has to fit into one of them.
Rung 4Glue
A filled block you can see at a glance belongs to a run, and that says how far it reaches.
Rung 5The whole line at once
One or two filled cells adrift in an open stretch, read with everything else on the line.
Rung 6Assumption
Fill a cell on spec, follow it until a line breaks, and rub it out again.

Common questions

Why does crossing a cell at the end of a line help?

Because it makes the line shorter. The same numbers in a shorter line have less slack, so the overlap arithmetic run again on what remains gives cells it could not give before.

What if the first cell of a line is filled?

Then the first run starts exactly there. Fill along to the length of that number and cross the cell straight after it — the run cannot reach any further than its own number.

Does a filled cell near the start count?

No. If anything between it and the edge is still undecided, the run might begin on either side of it. Using a block that is not at an end is a harder argument and lives two rungs up.