Whole-Line Analysis

When pairs and counting both run dry, stop looking at cells and start looking at arrangements. Two or three candidate endings for one line will usually agree about something.

List the endings, keep the agreement

Take a line that still has holes and write down every way it could legally be finished — every filling that respects the quota and creates no run of three. There are rarely many. Then compare them: any cell that holds the same symbol in all of them is settled, whichever ending turns out to be right.

That last move is the one to internalise. You are not deciding which ending is correct — you almost never can at this stage, and trying to is what turns the technique into guessing. You are extracting what the candidates have in common, which is knowledge you can write into the grid with complete confidence while the ending itself remains open.

It subsumes the cheaper techniques, which is why it sits above them rather than beside them. A double and a sandwich are both special cases of "every legal ending agrees here" — they are just the cases you can see without writing anything down. Line analysis is what you do when the agreement is real but no longer obvious.

Before — row 1 has four holes and two legal endings
	1	2	3	4	5	6
1	0					0
2					1
3					0
4					1
5	0
6		1

After — the two endings agree in two places
	1	2	3	4	5	6
1	0	1			1	0
2					1
3					0
4					1
5	0
6		1

Row 1 reads 0 _ _ _ _ 0. One more 0 and three 1s must go into the four holes. Putting that 0 in column 2 leaves 1 1 1 in columns 3 to 5; putting it in column 5 leaves 1 1 1 in columns 2 to 4; both are out. Only 0 1 0 1 1 0 and 0 1 1 0 1 0 survive — and they agree that columns 2 and 5 are both 1.

cells the reasoning read the cell it decides what it wrote

Choosing the line, and keeping the list short

Pick the line with the fewest holes, not the one nearest your last move. Four holes in a 6×6 line usually means two or three legal endings; six holes can mean a dozen, and a dozen candidates is a bookkeeping exercise rather than a deduction. If nothing has four holes or fewer, go back and sweep again first.

Build the list in order rather than trying to imagine it whole. Walk the line from one end, and at each hole ask what the quota and the no-three rule still permit given what you have written so far; most branches die within two cells. This is exactly what the solver on this site does, and it is why a line of fourteen with seven holes resolves in milliseconds instead of thrashing through thousands of combinations.

The technique is also self-limiting in a useful way. If a line produces only one legal ending, you have not done line analysis — you have found a line that counting would have finished anyway, and you spent more effort than you needed to.

What makes a puzzle hard, precisely

A hard grid here is one where the deduction solver, taking the cheapest available move every time, was at some point forced into this technique — no pair, no gap, no completed quota anywhere on the board. That is a strong statement: it means the grid contains a genuine bottleneck rather than merely being large.

It is also why size and difficulty are separate choices on this site. A 14×14 easy puzzle has a hundred and ninety-six cells and no bottleneck at all; a 6×6 hard puzzle has thirty-six cells and one. The second will take most players longer.

Where this sits on the ladder

Rung 1Double
Two alike side by side push the opposite symbol out to both ends.
Rung 2Sandwich
Two alike with one gap between them: the gap takes the other symbol.
Rung 3Counting
A line already holding its full share of one symbol has none left to give.
Rung 4Line analysis
List every way one line can be finished and keep what they agree on.
Rung 5Twin elimination
A finishing that would copy a finished line is not a finishing at all.
Rung 6Assumption
Put a symbol in, follow it until it breaks a rule, and write the other one.

Common questions

What is line analysis in a binary puzzle?

Listing every legal way one row or column could be completed, then writing in any cell that holds the same symbol in every one of those completions. The correct ending stays unknown; the agreement between them does not.

How do I keep the list of endings manageable?

Choose a line with four holes or fewer, and build the list left to right, dropping each branch as soon as it breaks the quota or makes three alike. Most branches die within two cells.

Is line analysis the same as guessing?

No. A guess commits to one ending and hopes; this commits to nothing and keeps only what every ending already agrees on. Nothing written in this way can ever need to be taken back.