Binary Puzzle Strategy: How to Solve Takuzu and Binairo
By the Sudoku Sagax editors · · 7 min read
The binary puzzle goes by many names: Takuzu, Binairo, Binoxxo or simply binary puzzle. The rules fit into three sentences, yet many players get stuck after the first easy puzzles. The reason is almost always the same: the simple patterns are well known, the fourth technique is not. Here are all four, in the order our own solver applies them.
The three rules in one paragraph
The grid is filled with zeros and ones. Every row and every column holds as many zeros as ones, which on our 8x8 grid means four of each. Three identical digits may never sit directly next to each other or on top of each other. And no two rows may be exactly the same, nor any two columns. The full version with examples is on our binary puzzle page.
There are no other rules, and every cell can be worked out from these. Our generator checks every puzzle before it is served: each one can be solved completely with the four techniques below, without trial and error at any point.
Techniques 1 and 2: pairs and gaps
The first technique follows directly from the rule against three identical digits. If two identical digits sit side by side, such as 1 1, the cells before and after them must hold the other digit, here a 0. Otherwise a forbidden run of three would appear. The second technique is the gap: if there is one empty cell between two identical digits, as in 1 _ 1, the middle gets the other digit.
These two patterns are the workhorses. In our puzzles they make up most of all steps on every level. If you are new to the puzzle, get into the habit of checking the neighbours right after every digit you place, because almost every new digit creates a new pair or closes a new gap.
One habit saves a lot of time: work line by line rather than cell by cell. Take one row, check every pair and gap in it, then move on to the next. After the rows come the columns. That way you miss less than when you jump around the whole grid.
Technique 3: counting
The third technique uses the balance rule. If a line already has its four ones, all remaining cells must be zeros, and the other way round. That sounds obvious, but it is easy to overlook because you do not keep count while looking for pairs.
It gets a little finer when only one digit is missing. If a row has three ones and three empty cells, exactly one of those cells must become a one. Pairs and gaps often rule out two of the three cells for that one, and the last cell is settled. Counting and pairs together solve many middle game situations where neither technique gets anywhere on its own.
A common mistake: thinking only horizontally
If you work row by row, it is easy to forget that every digit you place also counts in its column. A one that forms no pair in its row can sit right below another one in its column and create a pair there. So after every entry, check both directions: the neighbours to the left and right, and the neighbours above and below. It takes a second and prevents most of the wrong entries you would otherwise only notice much later.
Technique 4: line comparison
This is where most players get stuck. The fourth technique looks at one line as a whole and asks: which fillings are still possible for this line at all? On an 8x8 grid there are only 34 valid lines with four zeros, four ones and no three identical digits in a row. Once a line is half filled, usually only two or three of them are left. If they all agree on a cell, that cell is settled.
Take the row 0 1 0 _ _ _ _ 0 as an example. No pair and no gap helps, and counting does not settle anything yet. But try a 0 in the fourth cell in your head: then all four zeros of the row would be used, the three empty cells after it would all have to be ones, and that would be a forbidden run of three. So the fourth cell must be a 1. That is not guessing, it is a short proof.
The third rule is part of line comparison as well. If a row is filled except for two cells and one of the two options would make it a copy of a finished row, the other option must be right. This rule comes into play late, usually in the last few minutes, and that is exactly when people tend to forget it.
The way of thinking is the same as in nonograms, where you also check a single line against all of its possibilities. If you are good at one of the two puzzles, you have a head start on the other.
What separates the levels
Our three levels have on average 38, 24 and 16 clues out of 64 cells. More important than the number, though, is which technique you need. On Easy, three out of four puzzles can be solved with pairs, gaps and counting alone, without any line comparison. On Medium almost every puzzle needs it at least once. On Hard every puzzle needs it, and there roughly one step in five is a line comparison.
That gives a clear recommendation for practice. If you are comfortable on Easy and get stuck on Medium, you do not need more speed with pairs, you need line comparison. The best way to practise it is on lines that are already half filled: for each empty cell, ask yourself what would happen if the other digit stood there. With a little practice you see the answer without listing the possibilities.
This way of following an assumption to a contradiction and then discarding it is, incidentally, the same idea that helps with hard Sudokus. More on that is on the Sudoku solving techniques page, and an overview of all our puzzle types is on the puzzle types page.
Frequently asked questions about binary puzzle strategy
How do you solve binary puzzles?
With a fixed order: first pairs and gaps, then counting, then checking single lines for the fillings they still allow. After every digit you place, start again with the pairs.
Are Takuzu, Binairo and Binoxxo the same puzzle?
Yes, it is the same puzzle under different names. Some editions use X and O instead of 0 and 1, but the rules are the same.
Do you have to guess in a binary puzzle?
No. Every one of our binary puzzles can be solved completely with the four techniques. Line comparison is not guessing either, it is a short proof about a single line.
What should I do when I am stuck?
Recount the zeros and ones in every line, then look for a line that is half filled. Line comparison almost always finds a cell there.