Takuzu strategy: how to solve a binary puzzle

Technique 1 · basic

Pairs: block both ends

Two tiles of the same kind side by side are a pair, and a pair is a wall at both ends. A third tile of that kind next to it would make three in a row, so the tile just before the pair and the tile just after it both take the other kind. This is the step you will take most often in any Takuzu, Binairo or binary puzzle, and it works the same across and down. On a new board, scan every row and then every column for two alike touching. Each pair settles one or two tiles, and those new tiles often form a fresh pair or a gap of their own.

A pair against the edge of the grid blocks only one end, so it settles one tile, not two. Don't skip those: on a 6x6 board, more than half of all tiles touch the edge.

Where to look: two tiles touching in the middle of a line, where both ends are still open.

Technique 2 · basic

The sandwich: fill the gap

Two tiles of the same kind with one open tile between them make a sandwich. The middle tile must be the other kind, because the same kind there would close three in a row. It is the twin of the pair technique, and it is easier to miss, because the two tiles that matter don't touch. Train your eye on the patterns light, gap, light and dark, gap, dark, in every row and every column. Together, pairs and sandwiches cover most of the opening moves on an easy board, and they are still the first thing to check on a hard one.

A sandwich often hides in a line you think you are done with: one new tile can make a gap two tiles away from it. After every move, look two tiles out from it in all four directions.

Technique 3 · basic

Balance: count to half

Every row and column holds exactly as many light tiles as dark ones: three of each on a 6x6 board, four on 8x8, five on 10x10. So once a line has all the tiles of one kind it can take, every open tile left in it is the other kind. This is the balance rule, and it fills tiles in batches: two or three at once is common late in a solve. Counting is quick if you count only the kind that looks closer to done. A line with three dark tiles and two open ones is not finished until you notice it, and lines like that are what solvers overlook most when they feel stuck.

The count helps before a line is full, too. A 6x6 row with two dark tiles has room for just one more, and that single tile is what the next techniques are about.

Technique 4 · intermediate

Chains: switch between rows and columns

Every tile sits in one row and one column, so a tile you settle in a row is new information for its column. Strong solvers use that on purpose: after each move, they look at the crossing line before anything else. A count in a row places a tile, that tile makes a sandwich in its column, the new tile closes a gap in the next row, and so on. Following the chain is faster than scanning the whole board again, and it keeps your eyes where the board just changed, which is where the next step usually is.

When a chain stops, go back to a full scan for pairs, sandwiches and full counts. Only when that scan finds nothing is it time for the techniques below.

Technique 5 · intermediate

The last tile of a color

When a line needs only one more tile of a kind, that single tile has a job: it must break up every run of three open tiles, or those three would all become the other kind. So if the line has three open tiles side by side, the last tile of that kind goes somewhere inside them, and every open tile outside the run takes the other kind. Look for it in lines one short of half: two dark of three on a 6x6 board, three of four on 8x8. It is the step to try when pairs, sandwiches and counting run dry, and medium Evenly boards need this kind of line reasoning at least twice.

Check both kinds. A line one dark tile short is also a line several light tiles short, and the clue is in the kind with fewer tiles left to place.

Technique 6 · intermediate

Long stretches: how many fit

On 8x8 and 10x10 boards a line can have several open stretches, and the counting gets more interesting. The key fact: two open tiles next to a light tile can take at most one more light, or the three would be alike. The same goes for dark. So split the line into short stretches, work out the most light tiles each one can hold, and add them up. When the line needs more light tiles than the stretches can take, the rest must go in the open tiles outside them. This settles tiles that no pair, sandwich or count can reach, often far from the tiles that decide them.

It also works the other way: if the stretches can hold exactly as many light tiles as the line still needs, every open tile outside them is dark. When nothing moves on a big board, pick the line with the fewest open tiles and count its stretches.

Technique 7 · advanced

Unique rows and columns

Evenly keeps the fourth classic rule of Takuzu and Binairo: no two rows are the same, and no two columns are the same. Some binary puzzles leave it out, which is why this is the step people miss most. It matters near the end, when a line has two open tiles, one light and one dark to place, and both orders look legal. Compare the line with every finished line going the same way. If one order would make an exact copy of a finished row (or column), the other order is the answer. Hard Evenly boards are built so that you need this at least once.

Only finished lines count, and only lines in the same direction: a row is never compared with a column. With three or four open tiles the check still works, but list the legal ways to finish the line first and strike out the copies.

Technique 8 · advanced

Look ahead: test a tile

When none of the techniques above finds anything, pick an open tile and ask what would happen if it were light, then dark. Follow only the easy consequences: pairs, sandwiches and counts. If one choice runs into three in a row, or a line with too many of one kind, within a few steps, that choice is wrong and the tile is the other kind. This is not guessing: you never keep a tile you can't prove, and you take back everything you tried. Pick a tile where both lines through it are nearly full, so the consequences come fast and the chain stays short enough to follow in your head.

Evenly never requires it: every board can be finished with techniques 1 to 7. But looking ahead often finds a step sooner than a long count of stretches, and it is the tool to reach for on the hardest Takuzu puzzles in books and other apps.

Common mistakes

• Counting only one kind. A line with three light tiles is full of light, but a line with two dark tiles and a run of three open ones tells you just as much. Count both kinds in every line you look at.
• Forgetting the columns. Most people scan rows out of habit, because that is how they read. Half of all the steps on a board are in columns.
• Comparing unfinished lines. The no-copies rule only works against lines that are completely filled. Two half-done rows that look alike so far prove nothing.
• Guessing and carrying on. A wrong tile in a binary puzzle can survive many moves before it breaks a rule. If you test a tile, follow a few forced steps, then take everything back unless it ended in three in a row or a line off balance.