Binary Weave: turn a duplicate row into a precise deduction
List the two legal completions of 001…, then use an existing row to eliminate one while keeping column constraints in view.
Worked examples use the Standard board unless a different size is stated. Hard and Expert may change the board dimensions or amount of information to track. Teaching positions are explained separately from the playable board.
Separate three different tests
A Binary Weave row needs equal numbers of zeros and ones, no run of three identical symbols, and a completed pattern different from every other completed row. These tests do different jobs. Balance controls totals. Short-run rules control neighbours. Distinctness compares complete lines. Treating them as separate questions helps you explain why a particular square is forced instead of relying on the absence of a conflict highlight.
Use a separate six-square Standard-sized teaching row: 0, 0, 1, blank, blank, blank. Suppose another completed row already reads 0, 0, 1, 0, 1, 1. We are analysing these row facts, not presenting a complete starting board. The column entries are deliberately unspecified. Any row deduction must still be checked against the actual column evidence before it becomes a valid full-board conclusion.
Enumerate the small remainder
The partial row contains two zeros and one one. Its final three positions must therefore supply one zero and two ones. There are only three ways to place that single remaining zero: at position 4, 5 or 6. Writing all three is more reliable than assuming that the familiar-looking continuation is the only legal one.
If the zero goes in position 4, the completed row is 001011. If it goes in position 5, the result is 001101. If it goes in position 6, the result is 001110. All three have balanced totals. The last candidate, however, contains three consecutive ones at positions 3, 4 and 5. Reject it for that exact reason, rather than saying vaguely that it looks wrong.
| Candidate row | No three equal neighbours? | Compared with existing 001011 |
|---|---|---|
| 001011 | Yes | Duplicate: reject |
| 001101 | Yes | Different: retain locally |
| 001110 | No: positions 3–5 are 111 | Already rejected by the run rule |
Use the existing row only after local checks
We now have two completions that satisfy balance and the no-triples rule. One is an exact copy of the existing completed row 001011. Completed rows must be distinct, so that candidate is forbidden. The only remaining row pattern is 001101. In particular, position 4 must be 1, position 5 must be 0 and position 6 must be 1, conditional on the starting row facts being correct.
Without the existing completed row, the two candidates would remain possible. Balance and short-run rules alone cannot distinguish them. This is what makes distinctness useful reasoning rather than a final housekeeping check. It eliminates a complete proposed pattern, and the other constraints ensure that just one alternative remains. An incomplete row elsewhere is not automatically a duplicate just because its currently visible entries agree.
Look vertically before committing the row
Each newly inferred symbol belongs to a column too. If placing the one at position 4 would create a vertical triple or exceed that column’s allowance of ones, the row facts cannot all belong to a valid solution in that board. Do not keep the one merely because the row argument looked convincing. Revisit an earlier tentative entry, the assumed completed row, or a copied clue.
A row inference can be logically sound under its assumptions while exposing that one of those assumptions was wrong. This is different from claiming that distinctness has failed. The rule still rules out duplicate completed rows; the contradiction tells you the current collection of marks is inconsistent. Keeping track of which marks were fixed clues and which were experiments makes the backward check much easier.
Use a rejection notebook on larger boards
Hard rows have eight squares and need four of each symbol. Expert rows have ten and need five of each. Recompute the remaining counts for that size before listing candidate completions. You can often apply the same method to just two or three uncertain positions, without enumerating every possible full row. Record the exact reason a candidate is rejected: count, triple, duplicate or crossing-column conflict.
Rows are compared with rows, and columns with columns. A row matching a column is not a distinctness violation. Keyboard 0 and 1 let you enter the intended value directly; tapping a square cycles through its states. Undo restores an experimental entry, while Restart restores the clues. The aim is not to memorize 001101, but to recognize the moment when two locally legal patterns become one justified choice.
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