What is the Binary Puzzle uniqueness rule?
No two completed rows may contain the same sequence of 0s and 1s, and no two completed columns may contain the same sequence. Compare rows from left to right and columns from top to bottom. A match means every position is identical, not just that the lines contain equal numbers of each digit.
This guide uses the standard rules played on this site: balanced lines, no horizontal or vertical triples, and unique rows and columns. Uniqueness becomes useful when counting and checking pairs leave more than one possible completion. Instead of choosing between them, look for a completion that is already taken by another line.
The examples below are individual line deductions, not complete playable grids. They assume earlier entries are correct. After making a deduction, check the crossing lines as well.
How to use uniqueness with two empty cells
Look for a row with exactly two blanks and one missing 0 and one missing 1. There are only two ways to fill those blanks. If both arrangements satisfy the row’s balance and adjacency rules, a matching completed row can decide between them.
For example, a finished row reads 0 1 0 1 0 1. Elsewhere, an unfinished row reads 0 1 _ _ 0 1. Its four filled cells already contain two 0s and two 1s. The blanks must therefore be 0,1 or 1,0.
Putting 0 then 1 in the blanks would duplicate the finished row exactly. Putting 1 then 0 gives 0 1 1 0 0 1, which is balanced and contains no triple. The third cell is therefore 1 and the fourth is 0. No guess is needed.
0 1 _ _ 0 10 1 1 0 0 1The alternative completion, 0 1 0 1 0 1, is already used. The two blanks must take the other arrangement.
The same deduction works for unique columns
When horizontal scanning stalls, read the columns from top to bottom. Suppose a completed column is 0 0 1 0 1 1, while another column is 0 _ _ 0 1 1. The unfinished column needs one more 0 and one more 1.
Filling its second and third cells with 0 then 1 would copy the completed column. They must instead be 1 then 0, producing 0 1 0 0 1 1. Both candidate sequences satisfy the local count and no-three rules; the duplicate-column rule is what separates them.
The sequence below is written horizontally for readability, but each position represents a cell farther down the column. After filling the two cells, inspect their rows for new pairs and completed counts.
0 _ _ 0 1 10 1 0 0 1 1Compare columns with columns in the same top-to-bottom order. The two columns do not need to be adjacent.
Why three blanks may not give a forced move
Rejecting a duplicate does not always identify the remaining digits. Consider 0 _ _ _ 0 1. Three balanced, triple-free completions fit these clues: 0 0 1 1 0 1, 0 1 0 1 0 1, and 0 1 1 0 0 1.
If 0 1 0 1 0 1 is already a finished row, you can remove that candidate. But two candidates remain. They agree that the third cell is 1, so that cell is forced; they disagree at the second and fourth cells, so those positions must wait.
This is the broader method for using uniqueness when you are stuck in a Binary Puzzle: eliminate invalid full-line candidates, then place only values shared by every survivor. Another row or a crossing column may resolve the remaining ambiguity.
Common mistakes when checking duplicate rows
Two unfinished rows with the same visible pattern are not automatically duplicates. Their blanks may eventually differ. Compare possible completed sequences rather than treating matching blanks as matching digits.
A row may match a column. The standard uniqueness rule compares rows with other rows and columns with other columns; it does not prohibit a row-column match.
Reversed sequences are not automatically duplicates either. Compare positions in the same direction. Also, a line can differ from every other line and still be invalid because it has too many 0s or a run of three 1s. Keep all three rules active.
A short routine for finding the next move
First resolve touching pairs, separated pairs, and lines that already contain their full quota of one digit. Then compare nearly complete rows against finished rows, followed by nearly complete columns against finished columns.
When a duplicate candidate is removed, check whether one completion remains or whether several survivors share a value. Place only what is proven. Immediately inspect the crossing line, since one new digit may start a simpler chain of deductions.
If no candidate survives, recheck your earlier entries and counts. A contradiction at this stage can indicate a prior mistake; it is not permission to break the uniqueness rule.
Do duplicate rows have to be next to each other?
No. Two identical completed rows violate the rule wherever they appear in the grid. The same applies to columns. Compare against all finished lines of the same orientation, including ones far from the line you are solving.
Does every row with two blanks have a uniqueness move?
No. The two-blanks technique works when a possible completion would duplicate another row. If neither candidate duplicates a finished row, uniqueness alone may not decide the move. Check the crossing columns or return after more cells are solved.
Is line uniqueness the same as having one solution?
No. Line uniqueness is a rule about repeated rows and columns within a completed grid. Having one solution means only one complete grid fits all the starting clues and rules. A partially filled grid can obey line uniqueness and still admit more than one solution.
Practice the Binary Puzzle uniqueness rule
On your next board, find a nearly complete line and ask which finished lines it could copy. Write out the alternatives if needed. The Binary Puzzle uniqueness rule is most useful when you can name the exact duplicate being excluded and explain why the remaining placement follows.