How to Play Binairo
Binairo (also known as Takuzu or Binary Puzzle) combines binary logic with grid constraints. Fill the board with 0s and 1s following four simple rules. This guide teaches you everything you need to know.
What is Binairo?
Binairo is a logic puzzle played on a rectangular grid where every cell must contain either a 0 or a 1. The puzzle starts with some cells filled (givens) and your goal is to complete the grid. The constraints are deceptively simple: no three consecutive identical digits in any row or column, exactly half 0s and half 1s in each line, and all rows must be distinct from each other (as must all columns). These rules interact to create deep logical puzzles. Unlike puzzles requiring mathematical knowledge, Binairo relies on pattern recognition and constraint propagation. Each placement creates ripple effects through the grid. Every puzzle has exactly one solution. PencilGrid's generator guarantees uniqueness and ensures all puzzles are solvable through pure logic, with no guessing required.
The Four Rules
Core Constraints:
1.
No three consecutive. No three cells in a row or column may contain the same value (no 000 or 111).
2.
Equal count per line. Each row and column must have exactly N/2 zeros and N/2 ones (e.g., a 10×10 grid has 5 of each per line).
3.
Distinct rows. No two rows may be identical.
4.
Distinct columns. No two columns may be identical.
**Rule 1: No Three Consecutive** — The most immediate constraint. If you see two 0s next to each other with an empty cell adjacent (e.g., 00·), that empty cell must be a 1 to avoid three in a row. This works for columns as well. **Rule 2: Equal Count Per Line** — In an N×N grid, each row and column must contain exactly N/2 zeros and N/2 ones. For example, in a 10×10 grid, every line has exactly 5 of each. Once a line reaches 5 zeros, all remaining empty cells must be ones. **Rule 3: Distinct Rows** — No two rows may be identical. When a row nears completion, check it against already-complete rows. If filling a cell would create a duplicate row, the cell must take the opposite value. **Rule 4: Distinct Columns** — Same as above, but for columns. This constraint typically applies late in solving when columns are nearly complete.
Worked Example: 6×6 Binairo
The interactive example below shows a partially-filled 6×6 Binairo. We'll walk through two deductions to demonstrate how the rules combine to force specific placements.
Starting Grid:
· = empty cell | Highlighted cells show deductions
Deduction 1: Row 3, Column 3 must be 1
Rule applied: No three in a row
Column 3 already holds 0 at row 4 and 0 at row 5, directly below this cell. Placing 0 here would stack three 0s in the column, which is not allowed, so the cell must be 1.
Deduction 2: Row 2, Column 4 must be 1
Rule applied: Equal count per line
Column 4 already contains three 0s (rows 3, 5 and 6). Every line in a 6x6 grid holds exactly three 0s and three 1s, so the remaining empty cells in this column must all be 1.
Full Solution
Solution:
Verified unique solution
Essential Solving Techniques
**Complete Triples** — When two same values sit with an empty adjacent cell (00· or ·00 or 0·0), the empty must be the opposite value. **Balance Counting** — Count how many 0s and 1s are in each row and column. When a line reaches N/2 of one value, fill remaining empties with the other. **Forced by Exclusion** — If both the no-three rule and the balance rule eliminate the same value from a cell, it's forced to the opposite. **Pattern Matching for Distinctness** — As rows or columns near completion, compare them to finished lines. If one placement would create a duplicate, try the opposite value. **Iterative Constraint Propagation** — Each placement creates new constraints. After filling a cell, scan its row and column for new forced moves. Repeat until the grid is complete.
1. Complete Triples
If you see two same values with an empty cell adjacent (e.g., 00·), the empty must be the opposite value (1) to avoid three in a row.
2. Balance Counting
Count how many 0s and 1s are in each row/column. If a line already has N/2 of one value, all remaining empty cells must be the other value.
3. Pattern Recognition
As rows/columns near completion, check if they would duplicate an existing line. If placing a value would create a duplicate, try the opposite value.
4. Forced by Elimination
Combine rules: if no-three rule eliminates one value AND balance rule eliminates the same value, the cell is forced to the opposite.
Tips for Success
**Start with triples.** Look for two consecutive same values with an empty adjacent. These are the easiest forced moves. **Count aggressively.** Keep a running count of 0s and 1s per line. Mark lines that reach their limit visually or mentally. **Work rows and columns together.** A forced cell from a row constraint may trigger a column constraint (or vice versa). Alternate between perspectives. **Check distinctness incrementally.** Don't wait until the end. As lines near completion, compare them to finished lines. **Use the check feature.** Run a validation pass periodically to catch mistakes early. Finding an error after 50 moves is harder than finding one after 5.
Frequently Asked Questions
Can I guess in Binairo?
No guessing is needed. Every Binairo puzzle on PencilGrid has exactly one solution reachable through logic. The generator proves uniqueness. If you find yourself guessing, look for a simpler forced move you may have missed.
What does 'equal count per line' mean?
Each row and column must have exactly half 0s and half 1s. In a 10×10 grid, that's 5 zeros and 5 ones per line. In a 6×6 grid, it's 3 of each. Once a line reaches its limit for one symbol, all remaining empty cells must be the other symbol.
What's the best technique for beginners?
Start by looking for forced triples. If you see two identical adjacent values with an empty cell next to them (like 00·), the empty must be the opposite value (1 in this case) to avoid three in a row. This pattern is the easiest to spot.
How is Binairo different from Sudoku?
Binairo uses only two symbols (0 and 1) instead of nine digits. There are no boxes or regions. The constraints are different: Binairo focuses on preventing consecutive values, maintaining balance, and ensuring distinctness, while Sudoku focuses on uniqueness within rows, columns, and boxes.