Unique Rectangles

The Uniqueness Assumption in Sudoku

A properly constructed Sudoku puzzle has one and only one solution. If you can identify a candidate configuration that would create multiple solutions, that configuration is impossible. At least one candidate must be wrong, and you can eliminate it. The Unique Rectangle family applies this to four cells arranged in a rectangle spanning two rows, two columns, and exactly two blocks.

What Is a Deadly Pattern in Sudoku?

A deadly pattern is a configuration that would allow digit swapping to produce a second valid solution. Four cells forming a rectangle across exactly two blocks, all containing only {X, Y}, would create such ambiguity. Both arrangements satisfy all constraints. This is forbidden in a valid puzzle. The two-block requirement ensures the swap is invisible to all house types.

The Unique Rectangle Setup: Floor and Roof

Floor cells contain exactly {X, Y}. Roof cells contain {X, Y} plus extras. The extras are escape routes preventing the deadly pattern. The goal: prove that roof cells cannot both end up as just {X, Y}. Since the deadly pattern is forbidden, at least one roof cell must resolve to something other than X or Y.

Unique Rectangle Type 1: One Extra Corner

Three corners are bivalue {X, Y}. The fourth has {X, Y} plus extras. If the fourth were X or Y, the deadly pattern would complete. Eliminate both X and Y from the fourth corner. Level 6 (Hard).

Unique Rectangle Type 2: Shared Extra Candidate

Two floor cells {X, Y}. Two roof cells {X, Y, Z}. At least one roof must be Z. Eliminate Z from cells seeing both roof cells. Level 6 (Hard).

Unique Rectangle Type 3: Pseudo-Naked Subset

Two roof cells have different extras beyond {X, Y}. The extras form a pseudo-naked subset with other cells in the shared house. Level 7 (Very Hard).

Unique Rectangle Type 4: Conjugate Pair

One pair digit forms a conjugate pair between the two roof cells. This breaks the deadly pattern. Eliminate the other pair digit from both roof cells. Level 7 (Very Hard).

Unique Rectangle Type 5: Diagonal Extra

Extra candidate Z appears in diagonal corners or three of four corners. At least one Z-corner must be Z. Eliminate Z from cells seeing all Z-corners. Level 8 (Expert).

Avoidable Rectangle

Applies to rectangles where all corners are non-given cells (some already solved). The swap argument still applies because none were fixed by the constructor. Types 1-4 mirror standard UR types. Level 6 (Hard).

Hidden Unique Rectangle

In the roof cells' shared house, one pair digit appears as a candidate only in those two cells ("hidden"). That digit must go in one roof cell, breaking the deadly pattern. Eliminate the other pair digit from both roof cells. Level 8 (Expert).

Practical Tips for Finding Unique Rectangles

Start with bivalue cells. When you see two or more with the same pair, check for rectangle formation. Verify the two-block constraint. Check all orientations. Combine with other techniques after applying UR eliminations.

Summary

The Unique Rectangle family provides seven techniques exploiting the uniqueness assumption. They range from Type 1 (eliminate pair digits from the extra corner) to Hidden UR (find a hidden pair digit to eliminate the other). Mastering these significantly expands solving capability for Hard through Expert puzzles.