Unique Rectangle

A Unique Rectangle (usually shortened to UR) is a family of advanced Sudoku techniques built on one extra premise:

The puzzle being solved has exactly one solution.

The classic deadly rectangle uses four cells arranged in exactly:

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

If those four cells could all be restricted to the same pair of digits {X,Y}, the pair could be swapped around the rectangle to create two different completions that still satisfy the ordinary row, column, and box constraints.

A puzzle that is known to have one solution cannot reach that state.

Unique Rectangle techniques use that fact before the deadly pattern is completed. This Guide focuses on the foundation of the family and on Unique Rectangle Type 1, the cleanest place to learn the logic.

The uniqueness premise comes first

Classic Sudoku rules tell you what a completed grid must satisfy:

• each row contains digits 1–9 once;
• each column contains digits 1–9 once;
• each box contains digits 1–9 once.

Those constraints do not themselves say that the starting clues determine exactly one completion.

A well-constructed classic Sudoku is normally expected to be unique, but uniqueness is a property of the puzzle, not a hidden fourth row/column/box rule.

That distinction matters because a UR elimination can be perfectly consistent with the local rules and still be invalid if the puzzle has multiple solutions.

What is a deadly rectangle?

Imagine four unsolved cells at the corners of a rectangle. They occupy:

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

Suppose every corner contains only:

{4,9}

One possible assignment is:

4  9
9  4

The opposite assignment is:

9  4
4  9

In the four rectangle cells, both arrangements preserve the row and column restrictions. Because the geometry uses exactly two boxes, the box constraints are preserved too.

The two digits can therefore be exchanged without the basic Sudoku rules distinguishing the two completions.

That is the deadly pattern: a final state that would imply at least two solutions.

Why exactly two rows, two columns, and two boxes matter

The geometry is part of the proof.

A shape that merely looks rectangular is not enough.

Before using standard Unique Rectangle logic, verify:

2 rows
2 columns
2 boxes

The four corners will always occupy two rows and two columns by construction. The easy mistake is the box count.

A geometric rectangle can span four boxes. In that arrangement, swapping the two digits can change the box distribution and the classic deadly-pattern argument no longer follows automatically.

So the recognition order should be:

1. identify the four corners;
2. verify two rows;
3. verify two columns;
4. verify exactly two boxes;
5. only then inspect the candidates.

The two UR digits

A standard Unique Rectangle is built around two digits, often called the UR digits.

If the deadly state were completed, all four corners would be restricted to the same pair:

{X,Y}

In a real puzzle, a useful UR pattern often appears before every corner has reduced to exactly those two candidates.

That is why a Type 1 pattern can contain one corner with an additional candidate while the other three corners are clean {X,Y} cells.

The extra candidate is what prevents the deadly rectangle.

Unique Rectangle Type 1

A Unique Rectangle Type 1 has:

• the correct 2-row / 2-column / 2-box geometry;
• three corners containing only {X,Y};
• one corner containing {X,Y} plus one or more extra candidates.

For example:

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

Ask what would happen if D were allowed to use 2 or 7.

Then all four cells could collapse to the pair {2,7} and the deadly rectangle would exist. The puzzle would admit the interchangeable two-solution structure that uniqueness forbids.

Therefore D cannot be 2 or 7.

The UR digits are eliminated from D:

{2,7,9} → {9}

So:

D = 9.

If D contained several extra candidates, such as {2,7,9,5}, the conclusion would be:

remove 2 and 7 from D.

The extra candidates would remain for later logic.

Step-by-step Type 1 example

When reading a Type 1 example, verify the proof in this order:

1. confirm that the puzzle is known to be unique;
2. identify the four rectangle corners;
3. verify the 2-row / 2-column / 2-box geometry;
4. identify the two UR digits;
5. confirm that three corners are restricted to those two digits;
6. identify the extra candidate(s) in the fourth corner;
7. remove the UR digits from that fourth corner.

The elimination is not justified by the visual rectangle alone. It is justified by the rectangle plus the uniqueness premise.

Why Type 1 is an elimination before it is a placement

Type 1 is sometimes taught only through examples where the fourth corner has exactly one extra candidate and therefore becomes solved immediately.

That can hide the general logic.

The real conclusion is:

The fourth corner must use something outside the UR pair.

So the general move is:

remove the two UR digits from the fourth corner.

A placement happens only when that removal leaves one candidate.

This is the same useful distinction seen throughout advanced Sudoku:

• proof determines what can be eliminated;
• a Single may appear as a consequence.

Why this is not an ordinary contradiction

Suppose the fourth corner contains {2,7,9} and you temporarily imagine it as 2.

You may not immediately create:

• a duplicate 2;
• an empty candidate set;
• an invalid row;
• an invalid column;
• an invalid box.

Instead, the consequence is subtler: the grid can reach a state with another valid completion.

The contradiction is therefore not:

“2 breaks a basic Sudoku rule.”

It is:

“2 permits a non-unique completion, contradicting the verified unique-solution premise.”

This is why uniqueness techniques belong in their own conceptual family.

Unique Rectangle vs Naked Pair

A UR often contains repeated {X,Y} cells, so it can look like subset logic.

But the proofs are different.

Naked Pair

A Naked Pair uses:

• two cells;
• one house;
• exactly two reserved digits.

It is justified entirely by ordinary row/column/box constraints.

Unique Rectangle

A Unique Rectangle uses:

• four cells;
• two rows;
• two columns;
• exactly two boxes;
• a unique-solution premise.

Do not replace one proof with the other just because the candidate pairs look similar.

Unique Rectangle vs BUG+1

Both Unique Rectangle and BUG+1 are uniqueness techniques, but they detect different dangerous structures.

A UR is a local four-cell rectangle built around two interchangeable digits.

A Bivalue Universal Grave (BUG) is a much broader grid state in which unresolved cells and candidate occurrences form a binary structure capable of supporting multiple completions. BUG+1 uses one exceptional candidate to break that structure.

BUG+1 has its own VeyraPlay Guide and belongs to the same uniqueness family. Learning the UR premise first makes BUG reasoning easier to place in context.

Other Unique Rectangle types

Type 1 is only the beginning of the family.

Established uniqueness taxonomies include:

• Unique Rectangle Type 2;
• Type 3;
• Type 4;
• Type 5;
• Type 6;
• Hidden Rectangle;
• Avoidable Rectangle;
• BUG+1 as a related uniqueness technique.

Those patterns preserve the same central question:

What candidate structure would allow a forbidden non-unique state, and what must therefore be true or false to prevent it?

This page remains the canonical detailed explanation of Unique Rectangle Type 1. The broader Sudoku Uniqueness Techniques Guide maps Types 2–6, Hidden Rectangle, Avoidable Rectangle, BUG, and BUG+1. V2 deliberately does not create a separate URL for every UR subtype; those leaf pages remain a data-driven option if future search demand justifies them.

When should you look for a Unique Rectangle?

URs are not usually the first advanced pattern to scan for.

A practical time to look is when:

• candidate notation is accurate;
• the puzzle is verified unique;
• several bivalue cells repeat the same pair;
• those cells begin to align across two rows and two columns;
• easier local eliminations have already been exhausted.

A good trigger is seeing the same pair {X,Y} repeated in corners that almost form a rectangle.

Then check the box geometry before the candidates.

A safe Unique Rectangle recognition checklist

Before making any elimination, ask:

1. Is the puzzle verified to have one solution?
2. Do the four cells occupy exactly two rows?
3. Exactly two columns?
4. Exactly two boxes?
5. What are the two UR digits?
6. Does the candidate pattern match a UR type I actually know?
7. What exact candidate does the proof eliminate?

If any answer is uncertain, do not use the technique yet.

Common Unique Rectangle mistakes

Treating uniqueness as a hidden fourth rule

It is not. The technique needs an additional puzzle-level premise.

Ignoring the 2-box condition

A four-box rectangle is not the standard deadly rectangle.

Using UR on an unverified puzzle

If multiple solutions are allowed, avoiding a deadly pattern is no longer justified.

Seeing three {X,Y} cells and assuming Type 1 immediately

Candidate shape is not enough. Geometry and the fourth corner still matter.

Eliminating the extra candidate

Type 1 does the opposite: the UR digits are removed from the corner containing extras.

Memorizing type numbers without understanding the deadly state

If you cannot explain why the two solutions would exist, the pattern is not yet safe to use.

Recognition drill

Before looking closely at the digits, say:

“Two rows, two columns, two boxes?”

Then:

“What are the two UR digits?”

Then:

“Do three corners contain only those digits while the fourth contains extras?”

Finally:

“Is this puzzle verified unique?”

That order prevents most false positives.

FAQ

What is a Unique Rectangle in Sudoku?

It is a uniqueness pattern based on four cells in two rows, two columns, and exactly two boxes. The technique prevents candidate states that would allow the same two digits to swap and create multiple solutions.

Is Unique Rectangle valid on every Sudoku?

No. It assumes the puzzle is known to have exactly one solution.

Is uniqueness one of the basic Sudoku rules?

No. Rows, columns, and boxes define valid completions. Unique solvability is a property expected of a properly constructed classic puzzle.

Why must the rectangle occupy exactly two boxes?

Because the standard two-digit swap must preserve the box constraints as well as the row and column constraints.

What is Unique Rectangle Type 1?

Three corners contain only the two UR digits, while the fourth contains those digits plus extras. The two UR digits can be removed from the fourth corner.

Is a Unique Rectangle the same as a Naked Pair?

No. A Naked Pair is ordinary local subset logic. A Unique Rectangle depends on four-cell geometry and the unique-solution premise.

Should beginners use Unique Rectangles?

Not usually. Learn accurate candidates, subsets, Wings, and basic link logic first. Then UR becomes easier to recognize and justify correctly.

What to learn next

Read Unique Sudoku Solutions for the puzzle-level meaning of uniqueness and Sudoku Logic for the difference between ordinary constraints, contradiction, and additional premises.

Later uniqueness expansion can build from this page into other UR types and BUG+1 without changing this foundation.