Clearing a Row and a Column

The least clever thing on this ladder and the one that produces the most answers. It is also the one people forget, which is why their grids stop moving.

The whole of it, in a sentence

When a cell is down to one number, that number cannot appear anywhere else in its row or in its column, so cross it out of all of them. That is the entire technique. It applies to the numbers printed on the grid at the start just as much as to the ones you work out, which means the very first thing to do after reading the signs is to sweep every given number out of its two lines.

What makes it earn a rung rather than being called housekeeping is that it cascades. Crossing one value out of five cells will often leave one of those cells holding a single candidate, which is a settled cell, which sweeps its own row and column, which settles another. On an easy grid that chain reaction is the whole solve — the signs open the door and the sweep walks through the rest of the house.

It contains, without needing a separate name, the move a beginner meets first: a row with one empty cell takes the number the row is missing. That is this rule applied to the other n minus one cells of the row. There is no reason to teach it twice.

Before — the corner is down to a 1 or a 2, and there is already a 2 in the row
	1	2	3	4
1		4	2	3∨
2				2
3			1	4
4			3

After — the 2 goes, and the corner is settled
	1	2	3	4
1	1	4	2	3∨
2				2
3			1	4
4			3

The cell at row 1, column 3 holds a given 2, so no other cell in row 1 may hold one. The corner at row 1, column 1 was down to a 1 or a 2, and losing the 2 leaves it holding a single candidate — which makes it a settled cell, ready to sweep its own column in turn.

what the reasoning read what it decides what it wrote

Doing it in an order that does not waste your time

Sweep from whatever you settled most recently rather than starting at the top left each time. A newly settled cell is the only place new information has entered the grid, so it is the only place a fresh elimination can come from, and re-walking the whole grid to find that out is the difference between a two-minute puzzle and a ten-minute one.

Keep a mental note of lines that are nearly full. A row with two gaps left is two crossings-out away from being finished, and finishing a row is worth more than a random elimination elsewhere because it settles two cells and hands both of their columns something new. Working the nearly-finished lines first is how experienced solvers make a grid collapse rather than erode.

On paper, cross out rather than erase. A candidate you have eliminated and rubbed out cleanly looks identical to one you never considered, and the second time you reach that cell you will consider it again. The board here strikes them for you, which is the one genuine advantage a screen has over a pencil in this puzzle.

Why this plus the signs is exactly what easy means

An easy grid on this site is one the solver finishes using nothing but the sign chains and this sweep. That is the definition rather than an approximation of it: the solver always takes the cheapest available step, and if it never needed anything above the second rung then the grid is easy and is labelled so.

Which also means an easy grid is guaranteed to be finishable by a reader who knows only this page and the one before it. If yours has stalled, either a settled number has not been cleared out of its lines, or a chain of signs was read too shallowly on the opening pass. The first is far more common than the second.

Where this sits on the ladder

Rung 1Sign chains
A run of signs pins its ends: below k cells you cannot exceed n minus k.
Rung 2Singles
A number that is settled is gone from the rest of its row and its column.
Rung 3Only one home
A value that fits just one cell of a row or column belongs in that cell.
Rung 4Sign bounds
Read a sign against what its neighbour can still hold, not against the whole range.
Rung 5Pairs and sets
Two cells that own two values between them lock those values away from the rest.
Rung 6Assumption
Write a number in, follow it until the grid breaks, and rub it out again.

Common questions

What is the line sweep in futoshiki?

Taking a number that is settled and crossing it out of every other cell in its row and column. It applies to given numbers as much as to worked-out ones, and it is where most answers on an easy grid come from.

Why does my grid stop moving even though nothing is wrong?

Nearly always because something you settled has not been cleared from its two lines. The elimination is free, it cascades, and skipping it is the single commonest reason a futoshiki feels stuck.

Where should I sweep from?

From the cell you settled most recently. That is the only place new information has entered the grid, so it is the only place a new elimination can come from without re-reading everything.