This binairo solver fills in the binary puzzle you are working on. Pick the size, tap each cell that has a 0 or a 1, and press Solve for the full grid or Next step for one deduction at a time. It works the same for takuzu and binary puzzles; only the names differ.
The rules
- Every cell holds a 0 or a 1.
- No more than two of the same digit next to each other, across or down.
- Each row and each column has as many 0s as 1s.
- No two rows are the same, and no two columns are the same. Some puzzle books leave this rule out; untick it above the grid if yours does.
How to enter a puzzle
Tap a cell to select it, then tap it again to cycle through 0, 1 and empty, or use the buttons under the grid. On a keyboard, move with the arrow keys and type 0 or 1. You can also paste the grid as text, one line per row, with a dot for each empty cell.
How the solver works
The steps start with the three rules people use most. Two equal digits side by side force the opposite digit on both ends. A gap between two equal digits must be the other digit. A row or column that already has half its cells as 0s must be 1s everywhere else, and the same for 1s. When none of those apply, the solver looks at a whole row or column: it lists every way to fill it that keeps the rules (there are 34 valid ways to fill a row of 8, for example) and keeps only those that fit the digits already placed and do not copy a finished row. Any cell on which all those ways agree is decided. Every step on this page names the row or column it comes from.
To give the answer and prove it is the only one, the solver combines these rules with a search over the remaining cells. Its answers are checked against all four rules, and its solution counts for small grids are tested against a brute-force count of every possible grid.
Worked example
Suppose a row of 6 reads 0 0 . . 1 .. The two 0s side by side mean the third cell is a 1. Now the row has two 0s and two 1s, so one more of each is missing in cells four and six. If cell four were 1, the row would read 0 0 1 1 1, three 1s in a row, so cell four is 0 and cell six is 1.
Solving tips
- Scan every row and column for pairs and gaps first; they are the fastest wins.
- Count: once a row has its share of one digit, fill the rest with the other.
- Near the end, compare an almost-full row with the finished ones: the rule against equal rows often decides the last two cells.
For a picture puzzle with the same fill-or-leave-empty choice in every cell, try the nonogram solver; for stars instead of digits, star battle.
To play takuzu grids, open binairo online: new 6x6 to 10x10 puzzles and a daily grid, checked by this solver.
Solver tested . Found a problem? Tell us.
Questions
Is binairo the same as takuzu?
Yes. Binairo, takuzu, binary puzzle and binary sudoku are names for the same puzzle. Some versions use X and O or two colors instead of 0 and 1; just map one symbol to 0 and the other to 1.
Do rows really have to be different?
In the standard rules, yes. Some puzzle books drop that rule. Untick 'No two rows or columns the same' above the grid and the solver uses only the other rules.
What sizes does it solve?
Square grids with an even side from 4 by 4 to 16 by 16. All of them solve in a fraction of a second.
Why does the solver say there is no solution?
With the digits you entered, at least one row or column cannot be completed without three in a row, an unequal count, or a copy of another line. Check the digits you typed; one wrong cell is enough.
Can binairo be solved without guessing?
Well-made puzzles can. Next step shows the logical route; if the explained steps stop before the grid is full, the puzzle needs trial and error from that point, and the solver tells you so.