Binary puzzle rules
A Binary puzzle, also called Takuzu, Binairo or Binero, is a square grid to fill with 0s and 1s. There are three rules, and every row and every column must keep all three: it holds as many 0s as 1s, it never has three of the same digit side by side, and no two rows (and no two columns) are exactly the same.
The example is a real 6×6 puzzle from the game at Medium. It starts with 11 digits, and there is exactly one way to finish it: every row and column of the solution holds three 0s and three 1s, and you won't find 000 or 111 anywhere, across or down.
- Fill every empty cell with a 0 or a 1.
- Each row and each column holds as many 0s as 1s.
- Never three equal digits side by side, across or down.
- No two rows are the same, and no two columns are the same.
- A proper Binary puzzle has exactly one solution, and every puzzle here does.
How to play Binary online
Tap or click a cell to switch it: empty, 0, 1, then empty again. You can also select a cell and use the 0 and 1 buttons or keys. The digits you start with are darker and can't be changed.
By default, mistakes aren't marked as you play, so the thinking stays yours. Two switches under the board change that: “Mark broken rules” outlines three alike in a line, too many of one digit or two identical finished lines the moment they happen, and “Check against the answer as you play” marks every wrong digit at once, even one that breaks no rule yet. With either on, the Errors box counts the marked cells. Otherwise press Check whenever you want to know, and if the grid is full but not solved, the broken rules are marked for you.
Hint explains the next step instead of filling in a cell. It names the technique, highlights the cells it uses and says why in a sentence, for example “E4 and E5 are both 0. Three alike may never sit side by side, so E3 must be 1.” You can place it, or close it and make the move yourself. If a digit on the board is wrong, the hint says that first. Hints work at every level and in every language on the site.
Boards come in 6×6, 8×8, 10×10 (the default) and 12×12, at four levels. Symbols switches between 0/1, X/O and colours only; Line counters show how many of each symbol every row and column still needs. Undo and Redo go back and forth, Pause covers the board and stops the clock, and Reset and Solution ask before they do anything. Your game is saved in this browser: close the tab, and it is waiting when you come back.
| To do this | Mouse or touch | Keyboard |
|---|---|---|
| Fill a cell | Tap it: empty, 0, 1 | 0 or 1 (O or X) |
| Switch a cell | Tap it again | Space |
| Erase | Erase | Backspace, Delete |
| Get a hint | Hint | H, then Enter to apply |
| Undo, redo | Undo, Redo | Ctrl + Z, Ctrl + Y |
| Pause | The button by the timer | P |
| Move between cells | – | Arrow keys, after clicking a cell |
What Easy, Medium, Hard and Expert mean
Our levels are set by the techniques a puzzle needs, not by how many digits it starts with. Before you see a puzzle, the same engine that gives the hints solves it step by step, and the hardest step it needed decides the level. Every Medium needs at least one “last one”, every Hard a duplicate line or a whole-line step, and every Expert a “what if?”: the techniques below, in that order. The table shows what that means on the 10×10 board.
| Level | Clues | Easy steps | Needs beyond pairs, gaps and half lines |
|---|---|---|---|
| Easy | 34% | 100% | Nothing |
| Medium | 29% | 94% | A last one (100%) |
| Hard | 24% | 86% | A duplicate line (91%), line logic (33%) |
| Expert | 21% | 80% | A “what if?” (100%), a duplicate line (94%) |
Averages over 100 puzzles per level, 10×10, from the generator that runs this page, solved with the hint techniques. Clues: the share of the 100 cells filled at the start. Easy steps: the share of hint steps that are pairs, gaps or half lines.
Daily Binary, sharing, printing and your stats
The Daily puzzle is a 10×10 that everyone gets on the same day, new at midnight on your clock: Easy on Mondays, Medium on Tuesdays, Wednesdays and Sundays, Hard on Thursdays and Fridays, and Expert on Saturdays. Solve it each day to grow your days in a row; the puzzle you were playing waits until you come back to it.
Share copies a link that carries the whole puzzle, so a friend gets exactly the same grid, and the puzzle number under the board shows you are on the same one. Print makes a PDF of the puzzle you are playing, with its solution, or a pack of up to 50 new puzzles at the size and level you choose, one, two or four to a page. If you play with X and O, the sheets print X and O too. Every printed puzzle is checked again by a separate solution counter before it goes on the page.
Your stats count the puzzles you solve at each size and level, with your best and average times; only solves without hints set a best time. They are saved in this browser only.
How to solve Binary: pairs and gaps
Every solve starts with two patterns, and they place most of the digits: an Easy puzzle falls to them and the half-line count below, and even on Expert 80% of the hint steps use one of the three.
Two equal digits side by side push the other digit onto both ends. In row E, E4 and E5 are both 0. If E3 were 0 too, E3, E4 and E5 would be three 0s in a row, so E3 must be 1. The same pair makes E6 a 1 as well.
- 00 forces a 1 on both sides; 11 forces a 0 on both sides.
- 0_0 forces a 1 in the middle; 1_1 forces a 0.
- Scan every row, then every column: one new digit often makes a new pair.
Gaps: the same idea with a hole in the middle
When two equal digits have one empty cell between them, that cell takes the other digit. In column 2, B2 and D2 are both 0, so C2 must be 1: a 0 there would make B2, C2 and D2 three 0s.
Half reached: count every line
A row of six holds exactly three 0s and three 1s, a row of ten five of each. Once a line holds half its cells in one digit, every other cell in it is the other digit. Row A already has three 0s (A1, A2 and A4), so its empty cells are all 1: A5 here, and A6 as well.
Counting gets stronger as the board fills up, so check the count of every line you have just added to.
Last one: where the final digit can't go
Row D reads 1, 1, 0 and three empty cells, so it needs one more 1 and two more 0s. Try the 1 in each empty cell. In D4 or D5 the two 0s still fit, but if the 1 went in D6, the 0s would land in D4 and D5, right next to the 0 in D3: three 0s. When a line needs just one more of a digit, a spot for it that forces three of the other digit is ruled out. So the last 1 can't go in D6, and D6 is a 0.
This is the step every Medium puzzle needs (100% of them on the 10×10).
Line logic: every way to finish a line
Sometimes no single pattern decides a cell, but the whole line does. On this 8×8 board, row B reads _ 0 _ _ 1 _ _ 0: it still needs two 0s and three 1s. List the ways to place them without three alike side by side, and every one of them puts a 1 in B1. Only 33% of Hard puzzles need this step, but when nothing else moves, try it on the line with the fewest empty cells.
Duplicate lines: no two rows alike
Row C reads 1, _, 0, _, 1, 0: two empty cells, and it needs one more 0 and one more 1. There are only two ways to finish it, and one of them, 1 0 0 1 1 0, is exactly row F, which is already finished. Two rows may never be the same, so C2 is a 1 and C4 a 0.
This rule only bites when a line is nearly full, so compare each nearly finished line with the finished lines in the same direction. 91% of Hard puzzles need it.
What if? Following a chain
Expert puzzles need at least one step that no single line gives away. Pick a cell and suppose the digit it doesn't take, then follow the easy steps it forces. Suppose C2 were 0. Then C1 would be 1, A2 would be 1 and A1 would be 1, and with the 1 already in A3, row A would hold three 1s side by side. That breaks a rule, so C2 can't be 0: it is a 1.
The hint panel looks for the shortest chain it can find and names every step, so you can check it on the board. If a chain runs long, start it from a cell in a line that is nearly full.
How the puzzles are made and checked
The generator builds a full solution row by row from the lines that keep the rules, then takes digits away one at a time. A digit stays out only if the puzzle can still be solved step by step with the techniques of its level; otherwise it goes back. A puzzle those steps solve has exactly one solution, so you never need to guess, and the hardest technique it needed sets its level.
The daily puzzles, the first board you see and every printed puzzle are also checked by solution counters written separately from the game's engine.
| Line length | As many 0s as 1s | Also never three alike |
|---|---|---|
| 6 cells | 20 | 14 |
| 8 cells | 70 | 34 |
| 10 cells | 252 | 84 |
| 12 cells | 924 | 208 |
Every way to fill one line, counted by the game's engine. The no-three rule cuts the choices by two thirds on a 10×10, which is why counting pays off so early.
A solver and more two-symbol puzzles
Stuck on a Binary puzzle from a newspaper, a book or an app? Enter it in the Binary solver to check that it has one solution or to see the next logical step.
If you like logic with only two choices per cell, try Yin-Yang, where the black cells and the white cells must each form one connected group, or Magnets, where every magnet has a plus and a minus end and like poles may not touch. How every puzzle on this site is generated and tested is explained in how we make and test puzzles.