Binairo (Takuzu): the rules, and how to solve it without guessing

Guide written 4 October 2026

Binairo is a grid puzzle with two symbols and no arithmetic. You are never asked to guess: every well-made grid can be finished by logic alone. If you are new to it, the rules take a minute and the first technique takes another.

The rules

Fill every empty cell with one of two symbols. Below, they are X and O; in a printed puzzle they are usually black and white. Then check three things.

1. Balance. Every row and every column holds the same number of each symbol. On a 6×6 grid, that is three X and three O in every line.
2. No three in a row. You may not place three of the same symbol side by side, across or down.
3. No twins (classic Binairo only). No two rows may be identical, and no two columns.

Binairo and Takuzu are two names for the same puzzle. Grids are always an even size: 4, 6, 8, 10 and up.

Three deductions that do almost all the work

• Pairs. Two identical symbols side by side force the cells at both ends to be the opposite symbol, because a third would break rule 2.

  O

  X

  X

  O

  Start every puzzle by scanning for pairs. Along an edge, only one end exists, and it is forced too.
• Sandwiches. The same symbol on both sides of an empty cell forces the opposite in the middle. Put an X there and you would have three X.

  O

  X

  O

  .

  This is the same idea as a pair, with the gap in the middle. Look for it in every line.
• Counting. Once a row or column holds its full share of one symbol (half the cells), every empty cell left in it takes the other symbol.

  X

  X

  O

  O

  On a 4-wide line, two X means the rest must be O. Count again whenever you place something, because a line you ignored five minutes ago may now be one move from done.

A 4×4 puzzle, solved step by step

Here is a complete puzzle with four givens. Every move below is forced.

1. Pair. Row 3 starts X X, so the cell after the pair must be O.
2. Counting. Row 3 now has two X and one O, so its last cell is also O.
3. Sandwich. Column 1 reads X, empty, X: the empty cell (row 2) must be O.
4. Counting. Row 2 has two O (its first and last cells), so its two middle cells are X.
5. The rest follows the same way. The first column then holds X, O, X and needs one more O at the bottom, and the remaining cells fall out of counting and pairs.

The finished grid:

Check it against the rules: two X and two O in every row and column, no three together, and all four rows differ.

When a puzzle seems stuck

• Look at every line again, columns included. The usual cause of "no moves" is a column you stopped reading.
• Think in what would break. If an X in a cell would make three in a row or exceed the half-count, the cell is an O. Test both symbols and see which one fails.
• On big grids, finish the nearly full lines first. A line with one empty cell is a gift: its symbol is decided by the count.
• Twins help late. If a row has two cells left and another row already matches the rest, the twin rule tells you which way round they go.

Pondoku: a version with bridges

Pondoku is a Binairo-style game for phones, and it is a variant rather than the classic. It keeps the first two rules (equal counts, no three in a row) and swaps the third for little = and × signs between neighbouring cells: = means the two cells match, × means they differ. The symbols are a cross and a creature (a frog, cat, owl or bee skin). Boards are 4×4 to 10×10, and it has a daily puzzle. Its hint button only shows a move the rules force, so it will never ask you to guess. It is free with ads, with a one-time purchase to remove them, and it is on Google Play.