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Binairo Rules: How to Play Takuzu (With Examples)

Binairo is a logic puzzle where you fill a grid with two colours, here yellow and teal. Three rules decide every square: no three alike side by side, half of each colour in every line, and no two lines the same. It’s also called Takuzu, or the binary puzzle, when it’s played with 0s and 1s.

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The rules

Fill every square with yellow or teal, so that:

  1. No three alike are side by side, across or down. Two in a row is fine; three is not.
  2. Every row and every column is half and half. A 6×6 has three of each colour in every line.
  3. No two rows are the same, and no two columns.

Some squares are filled in at the start, as clues. When every square is filled and all three rules hold, the puzzle is solved.

What breaks a rule

Three alike side by side. Here are three teal in a row, which rule 1 doesn’t allow:

Three alike in a row: not allowed A row of six squares: yellow, then three teal side by side (ringed in red), then two yellow. Three alike next to each other breaks the first rule.
Three teal side by side (ringed): not allowed.

Too many of one colour. This row never has three alike together, but it has four teal in six squares. A row of six needs three of each:

Four teal in a row of six: not allowed A row of six squares: teal, teal, yellow, teal, teal, yellow. No three alike are side by side, but that's four teal, and a row of six needs three of each, so it breaks the second rule.
Four teal in a row of six (ringed): not allowed.

Two lines the same. Each of these rows keeps the first two rules on its own, but they’re identical, and rule 3 says no two rows may be:

Two rows the same: not allowed Two rows of six, both yellow, teal, yellow, yellow, teal, teal. Each keeps the first two rules on its own, but no two rows may be the same.
Two identical rows: not allowed.

In the game, a row or column that breaks a rule gets a red mark beside it, so you can see it straight away.

A puzzle from the game, solved

This is the very first level in the game: Easy level 1, a 6×6 with 13 clues. The clues sit on a grey square.

Easy level 1 from the game, before it's started A 6 by 6 Binairo with 13 clues. Row 1: yellow in columns 3 and 6. Row 2: yellow in columns 2 and 6, teal in column 5. Row 3: yellow in columns 4 and 5. Row 4: teal in column 2, yellow in column 3. Row 5: yellow in column 3. Row 6: yellow in columns 1, 2 and 5. 1 2 3 4 5 6 1 2 3 4 5 6
Easy level 1, before you start. Rows and columns are numbered from the top left.

It takes 18 steps, and every one of them uses one of three simple ideas. Here’s the first time each comes up.

Step 1: two alike, side by side

Row 3 has two yellow next to each other, in columns 4 and 5. A third yellow beside them would make three in a row, so the squares at both ends, columns 3 and 6, must be teal.

Step 1: a pair in row 3 Row 3 has two yellow side by side, in columns 4 and 5. The squares at either end of them, columns 3 and 6, must be teal. 1 2 3 4 5 6 1 2 3 4 5 6
Two yellow side by side in row 3: the squares at either end are teal.

Step 3: a line with half of one colour

Row 6 has yellow in columns 1, 2 and 5. That’s three yellow, half of its six squares, so it can’t have any more. Its other squares are teal.

Step 3: row 6 has its three yellow Row 6 has yellow in columns 1, 2 and 5: three, half of its six squares. Its other squares, columns 4 and 6, are teal. 1 2 3 4 5 6 1 2 3 4 5 6
Row 6 has its three yellow, so the rest of it is teal.

Step 5: a gap between two alike

Row 2 now has teal in columns 3 and 5, with one empty square between them. If that square were teal too, there would be three teal in a row, so it’s yellow.

Step 5: a gap in row 2 Row 2 has teal in columns 3 and 5, with one square between them. That square, column 4, must be yellow, or there would be three teal in a row. 1 2 3 4 5 6 1 2 3 4 5 6
One square between two teal: it must be yellow.

The rest

The same three ideas, applied one line at a time, finish the grid: 7 times two alike side by side, 3 times a gap, and 8 times a line with half of one colour. Every row and column ends up with three of each.

Easy level 1, solved The finished grid. Every row and column has three yellow and three teal, no three alike are side by side, and no two rows or columns are the same. 1 2 3 4 5 6 1 2 3 4 5 6
Easy level 1, solved.

Where to start in any puzzle

Look for two alike side by side, and two alike with one square between them. Both settle squares straight away, and every one of the game’s 150 levels has at least one to start from. On the clues alone, they settle about 5 squares on an Easy 6×6, 6 on a Medium 8×8, and 12 on a Hard 10×10. Each new square you fill can make a new pair or gap, so keep looking near the squares you’ve just filled.

Common mix-ups

  • Two alike is fine. Only three in a row breaks the rule. Two yellow side by side is allowed, and it’s a clue: the squares at either end are teal.
  • Half and half means exactly half. A row of 8 has four of each colour, not “about” four.
  • No repeats means rows with rows, columns with columns. A row may match a column; it just can’t match another row.

Next steps

Questions

Is Binairo the same as Takuzu?

Yes. Binairo, Takuzu and the binary puzzle are the same game under different names, and some versions use 0s and 1s instead of two colours. The three rules are the same everywhere.

Does every puzzle have one solution?

Yes. Every puzzle on PeerPuzzles has exactly one solution, and can be finished by logic alone, one row or column at a time, without guessing.

Why must no two rows be the same?

It’s the third rule of the puzzle, and it does real work: when a line has two squares left that could go either way, a finished line it mustn’t copy can decide them. Of the game’s 50 Medium levels, 41 need it at least once, and 38 of the 50 Hard levels.

What size should I start with?

6×6. Each row and column has only six squares, three of each colour, and the first moves are easy to spot.