Sector Chain
A sector is the pair of a cell's sides meeting at one of its corners, and the loop uses 0, 1, or 2 of them. A clue's opposite corners sum to it, so what travels is whether that count is odd or even — a 2 keeps it, a 1 or 3 flips it, and dots pass it on diagonally. A chain seeded at a border forces edges far from where it started.
How it works
- 1A SECTOR is the pair of a cell's sides meeting at one of its corners; the loop uses 0, 1 or 2 of them, and the =1 / ≠1 marks record whether you know that count to be odd or even.
- 2Two facts make odd-or-even travel. A clue's opposite sectors split its four sides, so their counts add to the clue — a 2 passes the parity along, while a 1 or a 3 flips it. And at the dot between two diagonal cells their sectors are all four of that dot's edges, so "0 or 2 lines" carries the parity across unchanged.
- 3The top-left 3 is the seed, and its corner is already spent: the two edges at the board's corner dot are drawn, so that sector holds 2 and the opposite one is left with exactly 1.
- 4Now run the diagonal. Each 2 turns "exactly one" into "exactly one" again, twice over, and each dot hop passes it straight through.
- 5The far 3 receives "exactly one" on its inner sector, so its other two lines have to go on the outer one: its bottom and right edges are both forced, three cells from where this started.
- Edge on the loop
- Edge the technique adds
- Cells that establish the pattern
- Corner constraint