Line Weave: filter complete placements before making the next mark
Combine an eight-square 2,2 clue with one filled square and one known empty square to find two surviving placements and four new forced marks.
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.
Known marks change the placement set
A line’s clue describes ordered filled runs, with at least one empty square between neighbouring runs. Overlap is useful, but comparing placements becomes more powerful after crossing lines have supplied additional filled or empty information. Keep only complete arrangements that agree with every justified mark. Then compare those surviving arrangements, rather than assuming a convenient run position.
Use a separate eight-square teaching line with clue 2,2. Number its squares 1 through 8. Suppose square 3 has been proved filled by its crossing column, and square 6 has been proved empty. These are assumptions for this line exercise, not a complete eight-by-eight puzzle. They must be justified in a real board before this reasoning is applied. We write 1 for filled, 0 for empty, and a dot for unresolved.
Count arrangements without losing the separator
The runs use four squares in total and their mandatory middle separator uses another one. An eight-square line therefore has three additional empty squares to distribute before the first run, inside the separating gap, or after the second run. Distributing those three indistinguishable extra spaces among those three locations gives ten placements. This counts possible line arrangements, not possible solutions of a whole board.
For a direct check, the ten strings are 11011000, 11001100, 11000110, 11000011, 01101100, 01100110, 01100011, 00110110, 00110011 and 00011011. Every string has exactly two runs of two. Inspecting the known third square removes the first four and the last string. Requiring the sixth square to be empty removes three more, leaving the two shown below.
| Surviving placement | Squares 1–4 | Squares 5–8 |
|---|---|---|
| First possibility | 0 1 1 0 | 0 0 1 1 |
| Second possibility | 0 0 1 1 | 0 0 1 1 |
| Agreement only | 0 · 1 · | 0 0 1 1 |
Mark agreement, not an entire favourite placement
Both surviving strings have square 1 empty, square 5 empty, and squares 7 and 8 filled. Those are four new deductions. Square 3 filled and square 6 empty were already known. Squares 2 and 4 remain unresolved: one is filled and the other is empty, but the line evidence does not choose which. The agreement pattern is 0·1·0011.
Do not fill square 2 merely because the first listed arrangement looks familiar. That would commit to one still-unproved candidate. Nor should both squares 2 and 4 be filled: the two alternatives are complete arrangements, not independent menus from which you can select preferred marks. Combining their optional filled squares would create an illegal three-square run through positions 2, 3 and 4.
Let one crossing deduction finish the line
Now suppose the column through square 2 separately proves that square empty. The first surviving placement, 01100011, is incompatible with that new fact. The second, 00110011, remains. Square 4 must therefore be filled, completing the first two-square run at positions 3 and 4. The final run occupies positions 7 and 8, separated by empty squares 5 and 6.
This extra empty mark must come from genuine column reasoning. Repeating an uncertain old cross as though it were independent evidence creates a circular argument. If the column instead proves square 2 filled, the other surviving row is selected. If it proves something that eliminates both arrangements, revisit the assumptions; the line cannot repair inconsistent input by inventing a third placement.
Use the same filter without an exhaustive notebook
On a larger line, writing every placement may be cumbersome. You can still test proposed run starts against known marks and keep track of which squares are common to all legal options. Preserve the clue order and mandatory separators throughout. A filled fragment might belong to more than one run, so do not attach it to a clue without checking the surrounding space.
In the game, crosses are reasoning notes; victory checks the filled runs in every row and column and does not require every unused square to be explicitly crossed. Use Mark empty or the X key for a proved gap, and Undo for an experimental mark. A locally completed line still needs its columns to agree. The useful result of this exercise is the filtered set and its shared information, not memorizing one eight-square answer.
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