Sector Parity
Hard
Used by Smart Hint
Each cell has four sectors, one per corner, and the number of them holding exactly one line is always even. Once three of a cell's four sectors are pinned to "exactly one" or "never exactly one", the fourth is settled — even when its own two sides are both still blank.
- Cells that establish the pattern
- Corner constraint
How it works
- Every cell has four sectors, one at each corner. Mark each one =1 ("exactly one of these two sides is a line") or ≠1 ("both, or neither") as you pin them down.
- The rule: a cell's four sectors always hold an EVEN number of =1's. Zero, two, or four — never one, never three.
- Why: an =1 sector means the loop separates the two cells sitting diagonally across it, and a ≠1 sector means it doesn't. Going around the cell, those four relationships form a closed ring — and a ring can't disagree with itself an odd number of times.
- Here three sectors are known: two =1 and one ≠1. That's two =1's so far, already even, so the last sector must be ≠1 — its two sides are both lines or both blank.
- The catch, and the whole point: this tells you nothing if you only look at the cell's own four sides — there the rule is automatically true and forces nothing. It pays off when the three known sectors were pinned by something ELSE (a dot's rule, a clue on the far side, a fact that travelled in along a chain of 2s) while the fourth sector's own two sides are still completely blank.