How to Solve a Binary Puzzle

One real 6×6, from nine clues to a finished grid. Every move below is the move the solver actually made, in the order it made it.

Where to look first, and why order matters

Start with pairs. Any two identical symbols already sitting side by side decide their own neighbours immediately, and finding them costs nothing but a sweep along each row and then down each column. They are the cheapest information on the board and they are usually the most plentiful at the start.

Then look for gaps between two matching symbols — a symbol, a hole, the same symbol again. The hole cannot repeat them without making three, so it takes the other. Same rule as the pairs, read backwards, and beginners routinely miss it for weeks because they are scanning for adjacency rather than for near-adjacency.

Only then start counting. Counting is slower — you have to tally a whole line rather than glance at two cells — so it is wasted effort while pairs remain. The rhythm that finishes puzzles is: sweep for pairs and gaps until nothing moves, count once, then go straight back to sweeping, because a count usually fills several cells and each of them creates new pairs.

The grid, move by move

The puzzle below starts with nine clues in thirty-six squares — a quarter of the grid — and is graded medium, which here means it needs counting somewhere but never anything harder. The four positions shown are the four moments where something new is being used; the twenty-one steps in between are more of the same.

Read each pair of grids left to right. The left one is the position as it stood, with the cells the reasoning read marked, and the cell it decides outlined. The right one is the same position a moment later.

1. Two 1s sit together at the end of row 2. The cell before them cannot be a third, so it is a 0 — one pair read, nothing else on the board consulted.

   The position

   	1	2	3	4	5	6
   1		1			1
   2					1	1
   3
   4	0		1		1
   5				0
   6						1

   A moment later

   	1	2	3	4	5	6
   1		1			1
   2				0	1	1
   3
   4	0		1		1
   5				0
   6						1

   Double

2. Row 4 now holds a 1, a hole and a 1. Filling that hole with a third 1 would break rule one, so it takes a 0. This is the shape people miss.

   The position

   	1	2	3	4	5	6
   1		1			1
   2				0	1	1
   3					0
   4	0		1		1
   5				0
   6						1

   A moment later

   	1	2	3	4	5	6
   1		1			1
   2				0	1	1
   3					0
   4	0		1	0	1
   5				0
   6						1

   Sandwich

3. Column 4 has acquired a pair of its own from the last two moves, and this time both ends are open — two cells from one observation.

   The position

   	1	2	3	4	5	6
   1		1			1
   2				0	1	1
   3					0
   4	0		1	0	1
   5				0
   6						1

   A moment later

   	1	2	3	4	5	6
   1		1			1
   2				0	1	1
   3				1	0
   4	0		1	0	1
   5				0
   6				1		1

   Double

4. Now the sweep runs dry, and counting takes over. Column 4 already holds its three 0s, so its last empty cell can only be a 1.

   The position

   	1	2	3	4	5	6
   1		1			1
   2				0	1	1
   3				1	0
   4	0		1	0	1
   5				0
   6				1	0	1

   A moment later

   	1	2	3	4	5	6
   1		1		1	1
   2				0	1	1
   3				1	0
   4	0		1	0	1
   5				0
   6				1	0	1

   Counting

cells the reasoning read the cell it decides what it wrote

Seventeen more moves of the same three ideas, and the grid closes. Here it is finished.

When you genuinely cannot move

Sweep again before concluding anything. The overwhelmingly common cause of a stuck grid at this level is a pair or a gap that was created by your own last move and has not been looked at since. Re-scan the two lines you most recently touched first — the information is nearly always there.

If that fails, count every line rather than the ones you suspect. A line that is one symbol short of its quota looks the same as a line that is three short until you actually tally it, and the tally is quick on six or eight cells.

If a full sweep and a full count both come up empty, the puzzle is above medium and wants a whole line analysed: pick the emptiest line you have and list the ways it could legally finish. That is the next technique up, and it is a page of its own. What the puzzle does not want, ever, is a guess — every grid on this site was checked to be finishable without one.

Common questions

Everything else on the site

• Daily puzzleA new grid every day
• Binary puzzleUnlimited grids
• The rulesThree rules, in full
• TechniquesThe whole ladder
• Doubles and sandwichesTechnique — easy
• Counting a lineTechnique — medium
• Line analysisTechnique — hard
• Twin eliminationTechnique — expert
• The assumptionTechnique — refused
• Printable puzzlesFor paper
• Daily archiveEvery past day