Sudoku Most Sudoku solving techniques rely on the rules of the game — each digit appears once per row, column, and box. Unique Rectangles take a different approach: they rely on the meta-rule that every properly constructed Sudoku has exactly one solution.
This uniqueness reasoning lets you spot deadly patterns — rectangular arrangements that would give the puzzle two solutions — and eliminate the candidates that would form them. It is one of the most satisfying and powerful intermediate-to-advanced techniques.
In this guide we cover what Unique Rectangles are, how the six types work, and walk through a real puzzle example with before-and-after diagrams.
✅ What Is a Unique Rectangle?
A Unique Rectangle (UR) is a group of four cells that:
- Occupy exactly two rows and two columns.
- Span exactly two boxes.
- Share the same two candidate digits (the “UR pair”).
If all four cells contained only those two digits, you could swap them and still have a valid grid — giving the puzzle two solutions. Because a valid Sudoku has exactly one solution, this arrangement (a deadly pattern) is impossible. At least one cell must contain an extra candidate, and the UR pair can be eliminated from that cell.
Unique Rectangles are the only mainstream technique that uses uniqueness reasoning rather than pure constraint logic. They work only on puzzles guaranteed to have a single solution — which includes all properly constructed Sudoku puzzles.
💀 The Deadly Pattern
A deadly pattern exists when four cells in a rectangle (2 rows × 2 columns × 2 boxes) each contain only the same two digits. The two digits can be swapped between the two possible arrangements without violating any Sudoku rule, producing two valid solutions.
Consider four cells with candidates {5,7} arranged like this:
- R7C5 = {5,7}, R7C7 = {5,7} — same row, different boxes
- R9C5 = {5,7}, R9C7 = {5,7} — same row, different boxes
Arrangement A: R7C5=5, R7C7=7, R9C5=7, R9C7=5
Arrangement B: R7C5=7, R7C7=5, R9C5=5, R9C7=7
Both satisfy every Sudoku constraint — two solutions. That is the deadly pattern.
🏠 Floor & Roof Cells
In the most common form (Type 1), three of the four rectangle cells already contain only the UR pair. These are the floor cells — they are locked into the pattern.
The fourth cell contains the UR pair plus one or more extra candidates. This is the roof cell. To prevent the deadly pattern, the roof cell must resolve to one of its extras — so the UR pair can be eliminated from it.
Floor cells: Only the UR pair — pattern is locked.
Roof cell: UR pair + extras — eliminate the UR pair, keep the extras.
🔎 Step-by-Step Example (Type 1)
Let’s walk through a real Unique Rectangle Type 1 for the digit pair {5, 7}.
Step 1: Spot the Rectangle
Four cells at R7C5, R7C7, R9C5, and R9C7 share candidates 5 and 7. They form a rectangle across two rows (7 and 9), two columns (5 and 7), and two boxes (Box 8 and Box 9).
Step 2: Identify Floor & Roof
- Floor: R7C5 = {5,7}, R7C7 = {5,7}, R9C7 = {5,7} — exactly the UR pair.
- Roof: R9C5 = {1,5,7} — the UR pair plus an extra candidate (1).
Step 3: Apply the Elimination
If R9C5 were 5 or 7, the deadly pattern would be complete — all four cells would contain only {5,7}, giving two solutions. Since the puzzle has a unique solution, R9C5 cannot be 5 or 7. Eliminate both:
- R9C5 — {1,
5,7} → {1} — a naked single!
Step 4: Result
R9C5 is now a naked single — it must be 1. This single placement can trigger further eliminations and simplify the rest of the puzzle.
Setup: Four cells in a rectangle (2 rows, 2 columns, 2 boxes) sharing two candidate digits.
Type 1: Three floor cells with only the UR pair, one roof cell with extras.
Elimination: Remove the UR pair from the roof cell.
Result: Fewer candidates; often a naked single.
🔄 All Six Types
| Type | Floor / Roof | Elimination Rule |
|---|---|---|
| Type 1 | 3 floor, 1 roof (with extras) | Remove UR pair from the roof cell |
| Type 2 | 2 floor, 2 roof on same side; both roofs share same extra digit x | Remove x from any cell that sees both roofs |
| Type 3 | 2 floor, 2 roof on same side; extras form a naked subset with neighbours | Apply the naked subset elimination |
| Type 4 | 2 floor, 2 roof on same side; strong link exists on one UR digit between roofs | Remove the other UR digit from both roofs |
| Type 5 | 2 floor (diagonal), 2 roof (diagonal); roofs share same extra | Same as Type 2 but diagonally |
| Type 6 | 2 floor (diagonal), 2 roof (diagonal); strong link on one UR digit | Same as Type 4 but diagonally |
Type 1 is by far the most common. Types 2–4 appear occasionally in hard puzzles, while Types 5–6 are rare.
🕵️ How to Spot Unique Rectangles
1. Scan for bivalue cells — cells with exactly two candidates.
2. When you find two or three bivalue cells sharing the same pair, check whether they form part of a rectangle across two boxes.
3. Look at the remaining corner(s). If they contain the UR pair plus extras, you have a Unique Rectangle.
4. Identify the type and apply the matching elimination rule.
Quick check: The rectangle must span exactly two boxes. If all four cells are in the same box, or they span three or four boxes, they cannot form a deadly pattern.
⚠️ Common Mistakes
1. Not Checking the Two-Box Requirement
The four cells must lie in exactly two boxes. A rectangle spanning three or four boxes cannot form a swappable deadly pattern because the box constraint would be violated.
2. Applying UR on Puzzles Without Uniqueness Guarantee
Unique Rectangles rely on the puzzle having exactly one solution. Do not apply this technique to puzzles from unknown sources or puzzle generators that do not guarantee uniqueness.
3. Confusing Floor and Roof Cells
Only remove the UR pair from the roof cell(s). The floor cells are already locked into the pair and must stay as they are.
4. Missing Higher Types
If two cells have extras (not just one), don’t dismiss the pattern. Check for Types 2–6 — they offer different elimination logic based on the extras and strong links.
📅 When to Look for Unique Rectangles
- Basic: Naked Singles, Hidden Singles, Full House.
- Intermediate: Naked Pairs, Hidden Pairs, Naked Triples, Pointing Pairs, Box/Line Reduction.
- Advanced: X-Wing, Skyscraper, Simple Colouring, Unique Rectangles.
- Multi-digit chains: XY-Wing, XYZ-Wing, W-Wing.
- Expert: Swordfish, Jellyfish, Avoidable Rectangles, BUG+1.
Puzzles requiring Unique Rectangles are typically rated Hard to Expert. Try our hard puzzles for practice.
🚀 Beyond Unique Rectangles
| Technique | What It Adds | Complexity |
|---|---|---|
| Unique Rectangles | Uniqueness reasoning on 2-digit rectangles | Advanced |
| Avoidable Rectangles | UR with solved cells already placed | Advanced |
| BUG+1 | Extends uniqueness to the entire grid (Bivalue Universal Grave) | Expert |
| Extended Unique Rectangles | Rectangles larger than 2×2 | Expert |
| Hidden Unique Rectangles | UR where the pair is hidden among other candidates | Expert |
Unique Rectangles are the entry point to a family of uniqueness-based techniques. Once you are comfortable spotting deadly patterns in 2×2 rectangles, you can extend the reasoning to Avoidable Rectangles (where some cells are already solved) and eventually to BUG+1 (which considers the entire candidate grid).
🎯 Practice Unique Rectangles
Sudoku Hard
Hard puzzles where Unique Rectangles and other advanced techniques are regularly needed.
▶ Play Hard SudokuSimple Colouring Guide
Another advanced single-digit technique using conjugate-pair chains.
▶ Read Simple Colouring GuideSudoku Solver
Enter your puzzle and see how the solver detects Unique Rectangles automatically.
▶ Open SolverFrequently Asked Questions
A Unique Rectangle exploits the one-solution guarantee. Four cells forming a rectangle across two boxes share two candidates. Since all four having only those two would allow two solutions (deadly pattern), extra candidates in at least one cell must survive, and the pair can be eliminated there.
A deadly pattern is when four cells in a rectangle (2 rows × 2 columns × 2 boxes) each have only the same two candidates. The digits could be swapped, giving two valid solutions. Unique Rectangles prevent this.
Floor cells contain only the UR pair — they are locked. The roof cell contains the UR pair plus extras. Eliminating the UR pair from the roof prevents the deadly pattern.
Six main types. Type 1 (one roof), Type 2 (two roofs, same extra), Type 3 (two roofs, naked subset), Type 4 (two roofs, strong link), Type 5 (diagonal, same extra), Type 6 (diagonal, strong link). Type 1 is the most common.
After basic and intermediate techniques are exhausted. Unique Rectangles sit at the advanced level. Look for bivalue cells sharing the same pair that could form a rectangle across two boxes.