At most two equal digits adjacent
Three of the same digit directly next to or below each other is never allowed. Two is fine — most deductions revolve around exactly that.
Binary puzzle rules and techniques
A binary puzzle has only three rules — and yet almost everyone gets stuck on their first grid. This page explains the rules precisely, followed by the five techniques that solve every grid step by step, without ever guessing.
The rules, precisely
The grid is always as wide as it is tall (for example 8×8) and every square eventually holds a 0 or a 1. The constraints:
Balance per row and column
Every complete row and every complete column contains exactly as many zeros as ones: four each on 8×8.
All rows and columns unique
No two complete rows are identical, and no two complete columns. You mostly need this rule near the end of a puzzle.
Five techniques that always work
With these five thinking steps — try them in this order — you can solve any binary puzzle:
1. Two of the same adjacent → fill the edges
See 1 1? Then a 0 must go on the left and on the right (otherwise you get three in a row). The same vertically. This is the fastest source of new digits.
2. Two of the same with a gap → the middle is opposite
See 0 _ 0? The middle must be a 1, because a 0 would make three in a row. Scan the grid for this pattern, horizontally and vertically.
3. Counting: is a digit used up?
Does a row of 8 already contain four ones? Then all empty squares in that row are zeros. Keep track per row and column of how many of each digit are already placed.
4. Compare nearly full rows
Is one row finished while another matches it up to two squares? The second row must not become identical — which pins down its last squares. The same holds for columns.
5. Elimination: try the wrong digit
If you are stuck, pick an empty square and think it through: what happens if this were a 0? If that inevitably breaks a rule, it is a 1. You will occasionally need this step on Expert.
Try it yourself
Theory only works once you apply it: play a binary puzzle at your own level, or print a set as a PDF and solve them on paper.
Frequently asked questions about the rules
Are two of the same digit next to each other allowed?
Yes, two is perfectly fine — only three or more directly adjacent is forbidden. Those pairs are your best friend: they force the surrounding squares.
Does every row need the same number of zeros as every column?
Every row and every column individually holds as many zeros as ones. On a 10×10 grid each row contains five zeros and five ones, and so does each column.
When do I need the uniqueness rule?
Usually near the end: when one row is finished and a nearly full row looks just like it, the second row must not end up identical. On smaller grids it rarely comes up; on 10×10 and 12×12 more often.
Is guessing ever necessary?
No. Our puzzles have exactly one solution and can be solved logically. If you are stuck, there is always a pair, a gap, a count or a row comparison you have not spotted yet — or use technique 5 and think one step ahead.