Binairo Puzzle & Rule Strength
Play the binary logic puzzle, and see which of its three rules actually does the work — the answer changes as the grid fills and again as the board grows.
Click a cell to cycle blank → 0 → 1 → blank. Given cells are shaded and cannot be changed. Everything runs in your browser; nothing is uploaded.
One of these three rules does nothing on its own — and finishes every puzzle
Binairo has three rules, and it is fair to ask which one you are actually using. We ran the solver with a single rule switched on at a time against 16 generated 10×10 puzzles and counted the blanks each could fill:
| Rule | Blanks it fills alone |
|---|---|
| No three the same in a row | 40.0% |
| Equal counts per line | 0.4% |
| No two lines identical | 0.0% |
The uniqueness rule fills nothing whatsoever — not a rounding-down of a small number, exactly zero cells. It needs a finished line to compare against, and a fresh puzzle has none, so it can never get started. On that evidence it looks like decoration.
It isn't. Leave the other two running and switch off only uniqueness: 14 of 16 puzzles stall, an average of 13.9 cells short of finished. The two rules doing all the visible work reach about 84% and stop dead, and the rule that does nothing alone is what carries the rest. Measuring rules one after another in sequence would have hidden that completely — isolation gives you the ranking, only combination gives you the interaction.
The strongest rule changes as you fill the grid in
Take a solved 8 × 8, blank cells at random, and vary how many. No rule is favoured by how the puzzle was built, because there is no puzzle — just a solution with holes in it:
| Blank | No three in a row | Equal counts | No duplicate lines |
|---|---|---|---|
| 15% | 72% | 95% | 51% |
| 25% | 65% | 75% | 53% |
| 35% | 56% | 46% | 37% |
| 50% | 41% | 16% | 10% |
| 65% | 25% | 3% | 2% |
| 80% | 7% | 0% | 0% |
On a nearly complete grid the counting rule is the strongest by a distance — 95% against 72%. On a sparse one it does nothing at all. The crossover sits at about 35% empty, which means the rule you should be leaning on genuinely changes partway through the puzzle in front of you.
The mechanism is that two of the three are whole-line rules. Counting needs a line to already hold half its symbols; uniqueness needs one finished outright. Both starve when lines are far from complete. The no-three rule only ever looks at three adjacent cells, so it doesn't care how much else is missing and degrades gently instead of falling off a cliff.
Measuring one board size would have given the ranking backwards
The same measurement, at half-blank, swept across board sizes:
| Board | No three in a row | Equal counts | No duplicate lines |
|---|---|---|---|
| 6 × 6 | 29.8% | 46.7% | 32.2% |
| 8 × 8 | 35.6% | 11.3% | 4.6% |
| 10 × 10 | 40.1% | 3.3% | 0.8% |
| 12 × 12 | 44.6% | 2.6% | 1.0% |
At 6 × 6 the counting rule is the strongest of the three and the no-three rule the weakest. At 12 × 12 that is exactly reversed, with counting down to roughly a seventeenth of its power. Anyone who measured a single board and reported a ranking would have published it upside down — which is why the sweep runs past the size where the pattern first looks settled.
Same cause as the fill-level inversion: at a fixed blank fraction a longer line sits further from complete in absolute terms, so the whole-line rules starve while the local one doesn't care how long the line is. The step-by-step figures wobble a little at this sample size, so the tests behind this page pin the endpoints and the change of leader rather than strict monotonicity — asserting the latter would be measuring the sample rather than the effect.
A quarter of the cells, whatever the size
Clues are stripped one at a time for as long as pure deduction can still recover the grid. What survives is almost exactly a quarter of the board, and the share barely moves:
| Board | Cells shown | Range |
|---|---|---|
| 6 × 6 | 26.0% | 22–31% |
| 8 × 8 | 25.4% | 23–27% |
| 10 × 10 | 25.5% | 22–28% |
| 12 × 12 | 25.2% | 22–27% |
A bigger Binairo is not a proportionally emptier one. And because the carving solver never guesses, a puzzle it can still finish is forced — so finishing is itself the proof that the answer is unique, and no separate solution count is needed. The exhaustive counts get computed anyway as a check on the machinery: 72 complete 4 × 4 grids and 4,140 at 6 × 6, the first of those confirmed by a second independent route that simply tests all 65,536 possible binary 4 × 4 grids one at a time and agrees exactly.
How to use
- Pick a board size and press New puzzle.
- Click a cell to cycle it between blank, zero and one.
- Keep every row and column half zeros and half ones, with no three the same in a row.
- No two rows may end up identical, and no two columns either.
- Check your grid, or reveal the answer if you are stuck.
Frequently asked questions
What are the rules of Binairo?
Three of them. No three of the same symbol may sit next to each other in any row or column. Every row and every column must hold equally many of each symbol. And no two rows may be identical, nor any two columns. The puzzle is also called Takuzu or Binary Sudoku, and every board is an even size so the equal-counts rule can be satisfied at all.
Which Binairo rule does the most work?
It depends on how full the grid is, which is the surprising part. On a grid that is only 15% blank the equal-counts rule settles 95% of what is missing against 72% for the no-three rule. On one that is 80% blank the counting rule settles essentially nothing and the no-three rule is the only one still working. The crossover sits at about a third empty.
Is the no-duplicate-lines rule worth anything?
On its own, nothing at all — measured across generated puzzles it fills exactly zero cells, because it needs a finished line to compare against and a fresh puzzle has none. But switch it off and leave the other two running and most puzzles stall well short of done. It is worth nothing alone and is still the rule that finishes the job.
Does board size change which rule matters?
Yes, and it reverses the order. At six by six the counting rule is the strongest of the three and the no-three rule the weakest. At twelve by twelve that is exactly backwards, with counting down to roughly a seventeenth of its power. Measuring a single board size would give you the ranking upside down.
How many cells does a Binairo puzzle show you?
About a quarter, and the share barely moves with size: 26% at six by six down to 25% at twelve by twelve. Clues are stripped one at a time for as long as pure deduction can still recover the grid, so a bigger board is not a proportionally emptier one.
Is every puzzle here solvable without guessing?
Yes, by construction. The generator only removes a clue if a solver that never guesses can still finish the grid, and a grid that pure deduction completes is forced. That makes finishing its own uniqueness proof, and it is separately confirmed by counting solutions on the smaller boards.
How many Binairo grids are there?
Seventy-two complete four by four grids and 4,140 at six by six, both counted exhaustively rather than estimated. The four by four figure is checked twice, once by the solver and once by simply testing all 65,536 possible binary grids one at a time.
Does this send anything anywhere?
No. Puzzles are generated, checked and solved entirely in your browser, and nothing is uploaded.
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