Binary Puzzle

Pick a size and a level and a fresh grid is built in your browser, checked, and put on screen. No limit, no account, nothing uploaded.

The three rules, in one paragraph

Fill every square with one of two symbols so that no three identical symbols sit side by side in any row or column; each row and each column ends up with an equal number of both; and no two rows are identical, nor any two columns. That is the whole game. Everything difficult about it comes out of those three sentences interacting, not out of anything else being added.

The puzzle is sold elsewhere as Takuzu and as Binairo, both of which are registered names belonging to their publishers. The rules are the same; only the branding differs. This site calls it a binary puzzle, which is what most English-speaking players type when they go looking for one.

Symbols are shown as 0 and 1 by default because that is the form the puzzle is usually named for and the form that reads fastest on a small screen. The two-colour view swaps them for a filled disc and a hollow ring — two shapes, not two hues, so the board stays readable with any kind of colour vision.

Where the difficulty actually comes from

Not from the clue count. That is the assumption almost every puzzle site is built on, and it does not survive contact with the puzzle: a grid with a third of its cells given can force reasoning that a grid with a quarter never needs. What decides how hard a binary puzzle is, is which deduction it makes unavoidable.

So the four levels here name techniques. Easy is finished with doubles and sandwiches — two cells you can take in at a glance. Medium adds counting, where a line that has already spent its share of one symbol has none left to give. Hard forces at least one line to be analysed whole, by listing the ways it could legally be finished. Expert is where the third rule does real work, ruling out a finishing only because some other line already uses it.

The grade is produced by a solver that works the way a person does, applying the cheapest technique that fits and moving on. The hardest technique it was forced to reach for is the level. That number is a property of the grid, not of the generator’s mood, and it is checked rather than estimated.

Every grid is proved before it is shown

A puzzle with two solutions is not a hard puzzle, it is a broken one, and a puzzle that can only be finished by guessing is worse — it wastes the reader’s time and then blames them for it. Both are ruled out here during generation rather than checked for afterwards.

Clues are carved out of a finished grid one at a time, and a clue is only allowed to stay missing if the technique solver can still complete the grid without it. Every cell reached that way was forced, which means the solution is unique by construction, and it means the puzzle is solvable at its advertised level because reasoning is literally what solved it. The independent counting solver — which knows nothing about techniques and simply searches — is run over generated puzzles in the test suite as a second opinion.

Nothing about this needs a network. The generator, the solver and the proof all run in your browser, in a worker thread so the page never stalls while a 14×14 expert grid is being built.

Common questions

Everything else on the site

• Daily puzzleA new grid every day
• The rulesThree rules, in full
• How to solveA 6×6, worked
• 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