The Unique Rectangle

A well-made Sudoku has exactly one solution. The Unique Rectangle uses that fact: it spots the start of a pattern that would give two solutions, and rules it out.

Ninesmith Sudoku · November 12, 2026

In short

Four cells on the corners of a rectangle, on two rows, two columns and exactly two boxes.
If all four held only the same two digits, the digits could swap and the puzzle would have two solutions.
So when three corners hold only those two digits, the fourth can't be either: strike both from it.

An example

A position from a real puzzle. Rows 3 and 7, columns 8 and 9, in boxes 3 and 9. Three corners hold only 4 and 7. The fourth, row 7, column 9, holds 4, 7 or 9. If it were 4 or 7, the 4s and 7s could swap, so it must be 9.

This one places a digit straight away: the fourth corner had only one other candidate.

Why it works

Picture the four corners holding just 4 and 7. Each row and each column would hold one 4 and one 7, and so would each of the two boxes. Swapping every 4 and 7 among the corners would break no rule, so there would be two solutions. A proper puzzle has one, so the deadly pattern can't happen, and the fourth corner must take something else.

The conditions

Exactly two boxes. Four boxes would not let the digits swap safely, and the logic fails.
Not given digits. The corners must be empty cells, not clues of the puzzle.
A unique puzzle. It only holds for a puzzle with one solution, which every proper Sudoku has.

This is the simplest form, type 1. Other types handle a fourth corner, or two, with different extra candidates.

Questions

What is a Unique Rectangle in Sudoku?

A technique based on the puzzle having one solution: four rectangle corners in two boxes can't all hold only the same two candidates. If three do, those two digits come out of the fourth.

Is using uniqueness cheating?

No. A proper Sudoku is defined as having one solution. The technique only uses that promise.

Next: all techniques · what to do when you're stuck

Drill the Unique Rectangle with the Smith

Ninesmith Sudoku is a calm Sudoku app for iPhone. Its Pattern Trainer drills the Unique Rectangle in sessions of 30 to 90 seconds, and in your own puzzles the Smith names the technique and shows the pattern before anything goes in.

Get the app

The full Unique Rectangle drills are part of Premium; the first scenario is free. No ads in any tier.