Binary Weave: combine counts with short runs

A zero is information too

Binary Weave uses only two symbols, but neither is a blank. A fixed zero is just as important as a fixed one and cannot be edited. The empty square is shown with a dot. Before reasoning about a line, distinguish the clues already present from the places that still need a decision. A six-square Standard row needs three zeros and three ones; its columns follow the same balance.

The three rules work together. Equal counts limit the total of each symbol. The no-triples rule controls neighbouring positions. Distinctness prevents one completed row from copying another completed row, and does the same for columns. Rows and columns are compared within their own groups: a row may match a column.

Solve a short run before doing arithmetic

Consider this separate Standard teaching row: 0, 0, blank, 1, blank, 1. The third square cannot be zero because that would make three consecutive zeros. Put a one there. The row now reads 0, 0, 1, 1, blank, 1. The fifth square must be zero to avoid a run of three ones through the third, fourth and fifth positions. The resulting row is 0, 0, 1, 1, 0, 1.

Now check the count: three zeros and three ones. That confirms the row rules, but it does not independently confirm the entries. Each new symbol also belongs to a column. If a proposed one would make a vertical triple or exceed the column's allowed total, an earlier assumption in your reasoning needs attention. A completed-looking row cannot overrule a conflicting column.

Use balance and distinctness as cross-checks

Balance sometimes resolves a line immediately. In a six-square row containing 1, 0, 1, 0, 1, blank, all three ones are already present. The remaining square must be zero. On an eight-square Hard line, the target is four of each symbol instead. Recount for the current board size rather than carrying the Standard limit into a larger puzzle.

When a row is nearly full, compare its proposed completed pattern with the other completed rows. An exact copy is forbidden even if its counts and short runs look valid. Do the corresponding comparison for columns. Treat this as a rejection test: discovering that a candidate duplicates a line does not excuse a different candidate from satisfying the other rules.

Separate a deduction from a test

A highlighted conflict identifies a present violation. The absence of highlighting does not prove a guess correct. If you test a value, remember what assumption you made and use Undo to return to it cleanly when a contradiction appears. Keyboard 0 and 1 set a value directly; tapping cycles through empty, zero and one. Restart restores the same clues. New puzzle changes a finite prepared layout through rotations, reflections or a global symbol swap, preserving its checked single solution without promising an unlimited collection.