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Unique Rectangle

Understand Unique Rectangles in Sudoku, the deadly pattern they avoid, and the solution-uniqueness assumption behind the technique.

A Unique Rectangle is an optional Sudoku technique based on a separate premise:

The puzzle has exactly one solution.

The standard deadly rectangle uses four cells that occupy:

  • exactly two rows;
  • exactly two columns;
  • exactly two boxes.

If all four cells can reduce to the same pair {X,Y}, the two digits can be swapped around the rectangle to create two valid completions.

A verified unique-solution puzzle cannot allow that final ambiguity.

Unique Rectangle techniques use this fact to eliminate candidates before the deadly pattern is reached.

Important: this is not a basic Sudoku rule

Classic Sudoku rules constrain:

  • rows;
  • columns;
  • boxes.

They do not state:

The starting clues must have exactly one completion.

Uniqueness is a property of the puzzle.

So VeyraPlay should only enable or teach uniqueness deductions when:

  • the puzzle is verified to have one solution;
  • the player has not disabled uniqueness techniques.

The deadly rectangle

Take four cells at the corners formed by:

  • rows 2 and 8;
  • columns 3 and 7;
  • exactly two boxes.

All four cells contain only:

{4,9}

One completion could be:

4  9
9  4

Another:

9  4
4  9

Both preserve the row/column/box restrictions for those four cells.

So the starting puzzle would not determine one unique final arrangement.

Why exactly two boxes matter

This is one of the most common mistakes.

Four cells can form a geometric rectangle across:

  • two rows;
  • two columns;
  • four boxes.

That does not create the same two-value swap under the box constraints.

For the standard Unique Rectangle geometry, verify:

2 rows
2 columns
2 boxes

before considering candidates.

Unique Rectangle Type 1

Type 1 is the cleanest introduction.

Three rectangle cells contain only:

{X,Y}

The fourth cell contains:

{X,Y} plus one or more extra candidates.

Example:

  • A {2,7}
  • B {2,7}
  • C {2,7}
  • D {2,7,9}

If D were 2 or 7, all four cells could collapse into the deadly {2,7} rectangle.

To preserve uniqueness, D must use an extra candidate.

So 2 and 7 can be removed from D.

If 9 is the only extra candidate:

D = 9.

Step-by-step Type 1 example

Why this is different from contradiction under ordinary rules

If you temporarily let the fourth cell take X, you may not create:

  • a duplicate digit;
  • an empty candidate set;
  • an immediate row/column/box violation.

Instead, you create another valid solution path.

The contradiction is with the unique-solution premise, not the basic local constraints.

That is why uniqueness must be explicit.

Other Unique Rectangle types

Established UR variants include several further types and Hidden Rectangles.

They use the same underlying idea but create different eliminations.

This Guide deliberately teaches:

  • deadly pattern;
  • geometry;
  • Type 1.

That is enough to establish the premise safely.

Additional UR types can be Expert expansion later.

Unique Rectangle vs Naked Pair

Repeated {X,Y} cells often look subset-like.

But Unique Rectangle logic depends on:

  • four-cell geometry;
  • exactly two boxes;
  • the unique-solution premise.

A Naked Pair uses only local unit constraints.

Do not substitute one proof for the other.

Product requirement for VeyraPlay

Uniqueness techniques create a real core/runtime requirement.

Before enabling a uniqueness hint:

solutionCountVerified === true
uniqueSolution === true

Conceptually.

The exact API can differ, but the semantic guarantee should not.

A manually authored training example also needs the same validation.

Common mistakes

Treating uniqueness as a hidden fourth rule

It is not.

Ignoring the 2-box condition

Geometry first.

Using uniqueness on an unverified puzzle

The elimination may be unjustified.

Assuming any {X,Y} rectangle is Type 1

Type 1 requires three clean UR cells and one corner with extra candidate(s).

Teaching advanced UR variants before the premise is understood

The assumption is more important than memorizing type numbers.

Recognition drill

Before candidates, say:

“2 rows, 2 columns, 2 boxes?”

Then:

“Do three corners reduce to the same two UR digits while the fourth carries extras?”

Then:

“Is this puzzle verified unique?”

Only after all three checks should Type 1 be considered.

FAQ

Is Unique Rectangle valid on every Sudoku?

No. It assumes the puzzle has exactly one solution.

Is uniqueness part of the standard row/column/box rules?

No.

Why two boxes?

That geometry permits the two UR digits to swap while preserving the box constraints.

Which UR type does this Guide teach?

Type 1.

What to learn next

Read Unique Sudoku Solutions for the puzzle-validity side, then use Advanced Mixed Practice to decide when uniqueness is appropriate versus when ordinary candidate logic is simpler.

theory · advancedUnique Sudoku SolutionsUnderstand what it means for a Sudoku puzzle to have exactly one solution, how software verifies uniqueness and why the property matters.strategies · advancedAdvanced Sudoku StrategiesLearn how advanced Sudoku solving shifts toward candidate patterns, Fish, Wings, Strong Links, Chains and disciplined search routines.concepts · advancedSudoku ChainsLearn how Sudoku Chains alternate Strong and Weak Links to propagate logical states, prove endpoint conclusions and eliminate candidates.