Unique Loop
A Unique Loop stretches the unique rectangle beyond four cells. The rectangle is the shortest deadly pattern; a unique loop is a longer, even cycle of cells that carry the same two candidates and would, if left unchecked, give the puzzle two solutions. Because a fair Sudoku has exactly one, the pattern must be broken — and that gives you your elimination.
The idea
Trace an even-length loop of cells, each holding the same two digits, that visits two cells in each box, row and column it touches. If every cell on the loop held only those two digits, you could swap the two digits all the way round the ring and get a second, equally valid grid — the deadly pattern. To prevent it, at least one cell on the loop must carry an extra candidate, and the true solution must use one of those extras. That lets you eliminate the two loop digits from a cell that has extras, or resolve a cell where the loop logic forces the issue, exactly as with a unique rectangle.
A worked example
Imagine six cells forming a loop — r1c1, r1c4, r2c4, r2c7, r3c7, r3c1 — all sharing {3,8}, exactly two in every row, column and box the ring touches. One of them, r1c1, also holds a 5. If the puzzle is to stay unique, that 5 cannot be a decoration — the solution must place 5 there. So 3 and 8 come out of that cell, breaking the deadly ring and often leaving a single behind. Every other cell of the loop stays {3,8} until further logic resolves it.
Six cells share {3,8} in an even loop — two per row, column and box. Pure pairs would swap round the ring for a second solution, so the extra 5 (top-left) must be the answer, clearing 3 and 8 there.
Step by step
- Find an even loop of cells sharing the same two candidates, two per unit it touches.
- Confirm the deadly pattern — swapping the two digits round the loop would give a second solution.
- Locate the extras — the cell or cells carrying a third candidate.
- Break the loop by forcing an extra digit, eliminating the two loop digits there.
Common mistakes
- Odd loops. The deadly pattern needs an even cycle with two cells in each unit — an odd loop does not swap cleanly.
- Variant grids. Killer cages and X-Sudoku diagonals can forbid the swap, so the argument only holds on standard, uniquely-solved grids.
- Ignoring the extras. The whole elimination lives in the cells with a third candidate — a loop of pure pairs is a warning the puzzle is flawed, not a technique.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.