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.

Works Classic Sudoku The theory behind every uniqueness technique on guaranteed-unique Classic grids. → Limited Killer Sudoku Extra cage constraints reshape which sets are unavoidable, so the plain theory is unsafe. → Limited Sudoku X The diagonals alter the unavoidable sets the standard argument assumes. → Limited Samurai Sudoku Best understood on a single standard grid. →

Where to practise this

This technique starts paying off at Evil, which is where puzzles first stop yielding to anything simpler. Reading about a pattern and spotting it on a live grid are different skills - the second one only comes from playing.

Play Evil Sudoku About Evil Sudoku

Related techniques

Unique Rectangle - Sudoku Uniqueness Technique Because a valid puzzle has only one solution, a four-cell rectangle sharing two candidates is forbidden - which forces an elimination. Jellyfish - the Order-4 Sudoku Fish One digit confined to the same four columns across four rows (or vice versa) - eliminate it from the rest of those columns. Finned Swordfish - Sudoku Fish Variation An almost-Swordfish with one extra candidate; the digit is eliminated only from cover cells that also see the fin. BUG+1 - Bi-Value Universal Grave Technique When all cells but one are bi-value, the extra candidate in that one cell must be the solution, or the puzzle would have two answers.