# Takuzu: how to solve the grid of 0s and 1s > Takuzu rules, or binairo, or binary puzzle, solving techniques with diagrams (pair, sandwich, counting, distinct lines) and all 14 possible 6 × 6 lines. - Web page: https://www.neurostim.fr/en/wiki/bitgrid - Play: https://www.neurostim.fr/en/bitgrid - En français: https://www.neurostim.fr/wiki/bitgrid.md - Updated: 2026-11-07 Takuzu, also known as binairo or the binary puzzle, is a grid you fill with 0s and 1s under three rules: as many 0s as 1s in each row and column, never three identical digits in a row, and no two rows or columns alike. It is all deduction, with no arithmetic. On Neurostim, [Bitgrid](https://www.neurostim.fr/en/bitgrid) offers a daily grid, the same for everyone, and a free run, at three difficulties. This guide explains the rules, the controls and the techniques, then shows which ones actually matter, using figures measured on 1,500 grids. ## In short - **Genre**: a single-player logic puzzle against the clock. - **Goal**: complete the grid of 0s and 1s while keeping the three rules. - **Sizes**: Easy 6 × 6, Medium 8 × 8, Hard 8 × 8 with fewer given cells and longer deductions. - **Daily grid**: everyone shares one per difficulty, and it changes at midnight, Paris time. Only the first try counts. - **Free run**: each game draws its own grid, and the leaderboard holds on to your best time that day. - **One solution**: every grid can be solved without ever guessing. - **Account**: optional, a guest name gets you ranked. - **Devices**: any screen, from a computer keyboard to a finger on a phone. ## The rules of takuzu ### The three rules 1. **As many 0s as 1s.** Each row and column holds exactly half 0s and half 1s: three of each on 6 × 6, four of each on 8 × 8. 2. **Never three in a row.** Three 0s or three 1s never touch, across or down. Diagonals have no limit. 3. **No duplicates.** No two rows are identical, and no two columns either. ### Edge cases - A row can match a column: rule 3 compares rows with rows and columns with columns, never one with the other. - Rule 3 only applies to full lines, but it also settles things earlier: a nearly finished line cannot end up like a line that is already full. - The cells given at the start cannot be changed. - A proper grid has only one solution: if you fill the whole grid without breaking a rule, that is the answer. ## Playing Bitgrid on Neurostim ### Controls | Action | Keyboard | Mouse or touch | |---|---|---| | Write a 0 | `0` | tap an empty cell | | Write a 1 | `1` | tap the cell again | | Cycle through empty, 0, 1 | `Space` | tap the cell | | Erase | `Delete` or `Backspace` | tap until it is empty again | | Move between cells | arrow keys | | | Send the grid | `Enter` | the "Submit the grid" button | The arrow keys skip the given cells. In a free run, `Tab` then `Enter` draws another grid, and `Esc` followed by `Enter` takes you out. These shortcuts do not exist on your first try at the daily grid. A practice try only has the exit. ### Automatic sending and errors Does a cell break a rule? It shows as an error at once: three identical digits, too many 0s or 1s on a line, two identical full lines. As soon as the grid is full and error-free, the page sends it by itself, without waiting for `Enter`. So the clock stops on your last cell. Your browser keeps your progress if you reload the page. ### The clock, the daily grid and the free run Time is counted on the server: it begins when the grid is shown, runs on even away from the page and only ends on an exact solution. Sending a wrong grid by hand brings no penalty beyond the seconds lost. Per difficulty, the daily grid accepts ten tries, of which only the first reaches the leaderboard. The free run allows up to 100 games a day. ### Leaderboards and rooms The daily grids have their [daily leaderboard](https://www.neurostim.fr/en/challenges/bitgrid/daily-easy), extended to the week and the month; [the speedrun FAQ](https://www.neurostim.fr/en/speedrun-faq) gives the scoring. In a [room](https://www.neurostim.fr/en/lobbies), one shared grid settles it between players: the first to complete it wins, the others can still finish, and the race runs for 5, 10 or 20 minutes at most by difficulty. ## Solving takuzu, from pairs to whole lines The first three techniques are enough for Easy and Medium. The others are for Hard. In the diagrams, the marked cell is the one the reasoning fills in. ### The pair Two identical digits side by side force the opposite digit on each side, otherwise you would get three in a row. ```puzzle . 1* 0 0 1* . ``` ### The sandwich Two identical digits with one cell between them force the opposite digit in that cell. ```puzzle . 0 1* 0 . . ``` ### Counting When a line already has its quota of one digit, everything else is the other. Here, on 6 cells, the three 1s are placed, so the three remaining cells are 0s. ```puzzle 1 0* 1 0* 1 0* ``` ### The last digit to place When only one copy of a digit is left to place, look at where it can go without creating three of the other. Here, on a line of 6 cells that starts and ends with 0, one 0 is left for four empty cells. If it went at either end of those four cells, the other three would be three 1s in a row. So it falls in the two middle cells, and the two marked cells are 1s. ```puzzle 0 1* . . 1* 0 ``` The same idea on 8 cells, with 0s in first, fourth and last position: one 0 is left for five empty cells. The three empty cells in a row on the right cannot be three 1s: the 0 is among them, and the two cells on the left are 1s. ```puzzle 0 1* 1* 0 . . . 0 ``` ### A line's possible endings The last digit is only a special case, and on Hard it is not always enough. The general method is the one the game's solver uses: list the ways to finish the line that keep rules 1 and 2, then keep the cells that have the same value in all of them. Here, on 8 cells, the line starts and ends with 0, with a 1 in fifth position. Only three endings work: 01011010, 01101010 and 01101100. All have a 1 in second position, so that cell is certain. ```puzzle 0 1* . . 1 . . 0 ``` ### Distinct lines When a line only has one 0 and one 1 left to place, compare it with the full lines. Here, the first row is 010110 and the second still has two empty cells in the middle: 01, then 10. It could end like the first: that is not allowed, so only the other order remains, 011010. ```puzzle 0 1 0 1 1 0 0 1 1* 0* 1 0 ``` The comparison also helps earlier: remove from the possible endings any that repeats a full line. Below, the second row would have three endings: 01101010, 10101010 and 10110010. The first is already taken by the top row. The other two start with 1 then 0, so those two cells are certain. ```puzzle 0 1 1 0 1 0 1 0 1* 0* 1 . . 0 1 0 ``` ### The 14 possible 6 × 6 lines Of the 64 strings of six 0s and 1s, only 14 keep rules 1 and 2, going by the list the game itself uses. They come in inverted pairs. On 8 × 8, there are 34 out of 256. | Line | Its inverse | |---|---| | 001011 | 110100 | | 001101 | 110010 | | 010011 | 101100 | | 010101 | 101010 | | 010110 | 101001 | | 011001 | 100110 | | 011010 | 100101 | Knowing this list helps you finish a line at a glance. For example, a line starting with 00 can only end in 1011 or 1101. ## Getting faster - **Do pairs and sandwiches first.** You can see them without counting, and on Easy and Medium they give more than six cells out of ten. - **Only count lines that are nearly finished.** A line of 8 with four 1s completes in one go. A half-empty line can wait. - **Read columns as much as rows.** A grid that looks stuck is often stuck in one direction only. - **On a keyboard, type the digits.** With a mouse, a 1 takes two clicks; `1` takes one. - **Do not look for rule 3 too early.** It is never needed on Easy or Medium. On Hard, it needs lines that are already full. - **Forget about `Enter`.** The grid sends itself as soon as it is full and error-free. To change grids without changing logic, [sudoku](https://www.neurostim.fr/en/wiki/sudoku) adds seven more digits and boxes, [nonograms](https://www.neurostim.fr/en/wiki/inkgrid) read blocks rather than digits, and [minesweeper](https://www.neurostim.fr/en/wiki/minesprint) has you deduce from neighbouring cells. ## Benchmarks: given cells and which rules matter ### What the generator leaves you Across 1,500 grids drawn by Bitgrid's generator in November 2026 (500 per difficulty), here is what you find at the start. | | Easy 6 × 6 | Medium 8 × 8 | Hard 8 × 8 | |---|---|---|---| | Cells given | 7 to 12 (9 in the middle) | 13 to 21 (16 in the middle) | 10 to 19 (14 in the middle) | | Share of the grid already filled | 25.8% | 25.5% | 22.7% | | Cells to fill, on average | 26.7 | 47.7 | 49.5 | | Grids finished by pairs, sandwiches and counting | 100% | 100% | 0% | | Grids that need the distinct-lines rule | 0% | 0% | 96.4% | So Easy and Medium never call for reasoning over whole lines. Hard always does, nearly always with rule 3. The generator removes cells as long as the techniques of the difficulty can finish the grid, so if you took away any given cell, those techniques would no longer be enough. ### What each technique brings Applying every possible pair and sandwich first, and counting only when they give nothing more, here is what fills the cells. | Technique | Easy | Medium | |---|---|---| | Pair | 40.3% | 44.3% | | Sandwich | 21.0% | 24.1% | | Counting | 38.6% | 31.6% | ### What a Hard grid asks of you Pairs, sandwiches and counting never finish a Hard grid, but they always give you a first move: across 500 grids, they always placed at least one cell at the start. They then stop with 34 empty cells at the median, out of about fifty to fill. You then have to reason over whole lines: four at the median (ten at most), plus three with the help of the distinct-lines rule (six at most). Only 3.6% of grids get by without rule 3. ### How many grids exist Counting every full grid that keeps the game's rules, we find 72 grids on 4 × 4, 4,140 on 6 × 6 and 4,111,116 on 8 × 8. Those numbers agree with sequence A253316 in the OEIS. That is very few next to sudoku, whose [guide](https://www.neurostim.fr/en/wiki/sudoku) gives the count. ### What about records? No world list of times could be verified for this game. Your real comparison is therefore [today's grid and its times](https://www.neurostim.fr/en/challenges/bitgrid/daily-easy). ## The history of takuzu The game goes by several names, some of them registered trademarks. English Wikipedia gives it two competing origins from about the same time, Tohu wa Vohu and Binairo; our sources do not date the first. - **Tohu wa Vohu**: the Italian Adolfo Zanellati invents the game under this name, meaning "formless and empty". - **2006**: the Takuzu trademark is registered in France on 31 January by the company Réalisations-Conceptions-Idées-Jeux, then in the European Union on 24 May 2013. - **2009**: Belgians Peter De Schepper and Frank Coussement create Binairo and register the trademark. - **2013**: the paper *Binary Puzzle is NP-complete*, posted on Nearly42, shows that deciding whether a grid has a solution is an NP-complete problem. - **2014**: Brian Kell opens sequence A253316 in the OEIS, counting the possible takuzu grids. - **2016**: Xavier Bultel and his co-authors publish a zero-knowledge proof for takuzu: proving you have the solution without showing it. Binero, for its part, is a trademark in France, and LinkedIn publishes a version of the game called Tango. Bitgrid is a Neurostim game: it takes the three common rules of the genre, under its own name, with a clock and a leaderboard. ## Frequently asked questions ### What is a binary puzzle? A binary puzzle, also called takuzu or binairo, is a logic puzzle in which you fill a grid with 0s and 1s under three rules: as many of each digit per row and column, never three identical digits in a row, and no two rows or columns alike. It takes no arithmetic and no luck. ### How do you play takuzu? Fill the empty cells with 0s and 1s, starting from the cells given. Look for two identical digits that touch or surround a cell, then for lines that already have their quota of one digit. On Bitgrid, type `0` or `1`, or tap the cell to cycle through the values. ### What are the rules of a binary puzzle? There are three: each row and column holds as many 0s as 1s, no digit appears three times in a row, and no two rows or two columns are identical. A row may match a column, however, and a proper puzzle allows only one solution. ### How do you solve binary puzzles? Apply the pair, the sandwich and counting in that order: on our Easy and Medium grids, those three techniques are always enough. When they run out, list the possible endings of the fullest lines and rule out any that repeat a full line. ### How do you solve hard binary puzzles? When the basic techniques stop, look for lines with only one copy of a digit left, because it has to break up any run of three empty cells, then list the possible endings of the others: whatever they share is certain. Finally, rule out endings that repeat a full line: 96.4% of our Hard grids need that. ### Is takuzu the same as binairo? Yes, the rules are the same: takuzu, binairo and binero are three names for one puzzle, registered as trademarks by different owners in France or the European Union. You will also find it as Tohu wa Vohu, Binary Puzzle or Tic-Tac-Logic, sometimes with symbols in place of digits. ### Is there a 10 × 10 takuzu? Yes, elsewhere, but not on Bitgrid, which offers 6 × 6 on Easy and 8 × 8 on Medium and Hard. Common grids run from 4 × 4 to 20 × 20 according to English Wikipedia, from 6 × 6 to 14 × 14 according to its French edition, usually with an even number of rows and columns. ### Is there a daily takuzu? Yes, Bitgrid puts out one daily grid per difficulty, shared by all players, and replaces it at midnight, Paris time. Only your first try reaches the leaderboard. To play more, the free run draws other grids, up to 100 games in a day. ### Is takuzu free to play online? Yes, no game of Bitgrid costs anything, and your browser is all you need, with no download or form to fill in. Your time is ranked under a guest name, and [the speedrun FAQ](https://www.neurostim.fr/en/speedrun-faq) says what an account is for if you want one later. ## Sources - Wikipedia, [Takuzu](https://en.wikipedia.org/wiki/Takuzu), checked on 7 November 2026. - Wikipedia (French), [Takuzu](https://fr.wikipedia.org/wiki/Takuzu), checked on 7 November 2026. - OEIS, [sequence A253316, number of takuzu grids](https://oeis.org/A253316), checked on 7 November 2026. - Nearly42, [Binary Puzzle is NP-complete](https://www.nearly42.org/cstheory/binary-puzzle-np-complete/), 27 March 2013. - Xavier Bultel, Jannik Dreier, Jean-Guillaume Dumas, Pascal Lafourcade, [Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen](https://arxiv.org/abs/1606.01045), arXiv, 2016. --- Neurostim, free daily brain games since 2008: https://www.neurostim.fr/en · Wiki: https://www.neurostim.fr/en/wiki · llms.txt: https://www.neurostim.fr/llms.txt