The Double and the Sandwich

Both come out of rule one alone, both read two cells and decide a third, and between them they finish every easy puzzle on this site.

The double: two alike, side by side

When two identical symbols already sit next to each other, the cells immediately outside them cannot repeat the symbol — a third would make three in a row. So both ends flip, and a pair you can see in one glance places up to two cells at once.

It works in both directions and at the edges. A pair at the very edge of a line has only one outside cell, and that cell is still decided; a pair in the middle decides two. The only case that yields nothing is a pair already bracketed by filled cells, which is why the sweep is worth repeating rather than doing once.

This is the single most productive move in the puzzle by a wide margin. On a 14×14 easy grid it accounts for most of the placements outright, and the ones it does not make it usually sets up.

Before — row 3 holds a pair
	1	2	3	4	5	6
1
2				0
3		1	1
4	0
5		1
6

After — both ends are forced
	1	2	3	4	5	6
1
2				0
3	0	1	1	0
4	0
5		1
6

Row 3 already holds two 1s side by side in columns 2 and 3. A 1 on either side of them would make three, so both the cell before and the cell after take a 0 — two cells placed from one glance, with nothing else on the board consulted.

cells the reasoning read the cell it decides what it wrote

The sandwich: two alike with a gap

The same rule read backwards. If a symbol, an empty cell, and the same symbol again sit in consecutive positions, the empty cell cannot hold that symbol — it would complete a run of three — so it takes the other one.

This one is missed far more often than the double, and the reason is worth naming: people scan for adjacency. Two matching symbols with a hole between them do not look like a pattern, so the eye skips them, and a grid that is genuinely still moving feels stuck. It is the commonest cause of a beginner abandoning an easy puzzle.

A deliberate second pass fixes it. Sweep once for touching pairs and once for gapped pairs, along the rows and then down the columns, and treat them as two separate errands rather than one — you are looking for different shapes and you will not find both at once.

Before — column 5 has a gap between two 1s
	1	2	3	4	5	6
1	0					0
2					1
3
4					1
5	0
6		1

After — the gap takes the other symbol
	1	2	3	4	5	6
1	0					0
2					1
3					0
4					1
5	0
6		1

Column 5 holds a 1 in row 2 and a 1 in row 4, with row 3 empty between them. Filling that hole with a 1 would make a vertical run of three, so it must be a 0. Nothing about the rest of the column matters — this is two cells deciding a third.

Why both are called cheap

Neither one needs you to know anything about the rest of the line. No count, no tally, no memory of what is elsewhere in the grid — two cells in, one or two cells out. That locality is what makes them fast to spot and safe to trust when you are tired.

It also makes them the right first move every single time. Because they read so little, they cost almost nothing to try, and because every symbol they place creates new neighbours, they tend to cascade: one pair resolves, the cell it decides forms a new pair with what was already beside it, and that resolves too. A single sweep after a counting move often unwinds five or six cells this way.

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 a double in a binary puzzle?

Two identical symbols already side by side. Because a third alike would break the no-three rule, the cells immediately outside the pair must both take the other symbol.

Why do I keep missing sandwiches?

Because the eye scans for adjacency and a gapped pair does not look adjacent. Make it a separate deliberate pass — first touching pairs, then symbol-gap-symbol — rather than trying to spot both in one sweep.

Can these two techniques finish a whole puzzle?

On an easy grid here, yes — that is precisely what "easy" is defined to mean on this site. From medium upward they will stall at some point and counting takes over.