Peak Sudoku

Unique Rectangle

A pattern that would create two solutions – logic forces it to break.
All techniques

Overview

Tier 4
Fish, wings, chains & uniqueness
Typical difficulty
Peak Score 55–70
Also known as
UR, Deadly Pattern, Uniqueness Test

Because every proper Sudoku has exactly one unique solution, four cells forming a rectangle across two rows, two columns, and two boxes cannot all share the exact same two candidates. Logic forces the pattern to break.

The Unique Rectangle technique leverages a fundamental premise of Sudoku: every well-formed puzzle has a single unique solution. If four cells forming a rectangle across two rows, two columns, and exactly two 3×3 boxes were all to contain only the same two candidates, the puzzle would have two valid solutions (swapping the numbers in opposite corners). Since a valid puzzle forbids this 'deadly pattern', any candidate that would force the deadly pattern must be eliminated.

Worked examples

1 / 5
Reading the exampleCells forming the patternCells losing candidatesEliminated candidates
R3C1, R3C2, R9C1 and R9C2 form a rectangle in two boxes, and three corners hold only {5,6}. If R3C2 were also just {5,6}, the corners could swap their 5 and 6 for a second solution. The puzzle has only one, so 5 and 6 can be removed from R3C2.
Removes: 5 from R3C2 · 6 from R3C2

How to apply it

Spotting the pattern and making the deduction, in order.

  1. Find four cells forming a rectangle across two rows, two columns, and exactly two 3×3 boxes.
  2. Confirm that three of the corners are bivalue cells with the exact same two candidates (for example, 4 and 9).
  3. Notice that the fourth corner contains 4 and 9 plus one or more extra candidate numbers.
  4. If that fourth corner were to take 4 or 9, all four corners would be locked on {4, 9}, creating an invalid puzzle with two solutions.
  5. Because every proper Sudoku has only one unique solution, the deadly pattern cannot be allowed to form.
  6. Eliminate 4 and 9 from that fourth corner cell, leaving behind only its extra candidates.

Common mistakes

  • Attempting a Unique Rectangle where the four cells sit in four different boxes. The cells MUST sit in exactly two boxes.
  • Applying uniqueness on puzzles that might not have a guaranteed unique solution (though all Peak Sudoku puzzles are mathematically guaranteed unique).
  • Confusing Type 1 (direct elimination) with Type 2 or Type 4.

Frequently asked questions

Why must the cells span exactly two boxes?
If the rectangle spanned four boxes, placed digits in the boxes could differentiate the solutions. Only when the four cells span exactly two boxes does the deadly interchangeable pattern occur.
Is using uniqueness considered 'pure' Sudoku logic?
Yes! While some purists prefer constructive proofs, uniqueness deductions are 100% mathematically sound for all valid puzzles published with a single solution.
Peak Score 65–75BUG+1All unsolved cells are bivalue except one – that cell's odd digit is forced.Peak Score 30–45Naked PairTwo cells in the same house contain exactly the same two candidates.

Put Unique Rectangle into practice

Try a puzzle from the Hard tier (Peak Score 51–60).

Play a Hard puzzleAll 25 techniques