Unavoidable Sets
Unavoidable sets are the theory beneath every uniqueness technique — the unique rectangle, the avoidable rectangle and the unique loop are all special cases. Understanding them explains why those shortcuts are safe on a proper puzzle.
The idea
An unavoidable set is a group of solved cells whose digits could be rearranged into a different but equally legal completion of the grid. The four corners of a deadly rectangle are the smallest example: swap the two digits and every row, column and box is still satisfied. Because such a set has two valid arrangements, a puzzle can only have a single solution if it pins down at least one cell of every unavoidable set with a given clue. Turn that around while solving: if the givens and your placements would leave an unavoidable set free to flip, the puzzle would have two solutions — so whatever you are considering must be wrong, and you can eliminate it.
A worked example
Four cells forming a rectangle in two boxes — r2c2, r2c7, r6c2, r6c7 — all reduce to the pair {2,7}. Those four cells are an unavoidable set: placing 2 and 7 clockwise, or the other way round, both satisfy every row, column and box. Since the puzzle is unique, one of the four must actually be something other than 2 or 7 — there has to be a clue or forced digit breaking the set. That realisation is exactly the unique-rectangle elimination, seen from the theory that justifies it.
The four corners all reduce to {2,7} across two boxes — an unavoidable set. The 2s and 7s can sit on either diagonal (dashed) and both fillings are legal, so a unique puzzle must pin at least one corner from outside.
Step by step
- Spot a candidate unavoidable set — cells whose digits could be permuted into a second solution.
- Check it is truly free — the swap must break no row, column or box rule.
- Invoke uniqueness — a single-solution puzzle must fix one cell of the set.
- Eliminate any placement that would leave the set able to flip.
Common mistakes
- Not actually unavoidable. If the swap breaks any rule, the set is fine and no uniqueness conclusion follows.
- Variant grids. Cages and diagonals change which sets are unavoidable, so the plain theory does not carry to Killer or X-Sudoku.
- Using it on unknown puzzles. The argument assumes a guaranteed single solution — never apply it where that is not promised.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.