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Module 6 · Links & ChainsLesson 27 of 28

Unique Rectangles

Understand the unique-solution assumption, identify the four-cell deadly rectangle geometry, and apply Unique Rectangle Type 1.

Outcome

You can state the unique-solution assumption, verify the 2-row / 2-column / 2-box deadly geometry, and apply Unique Rectangle Type 1 only when the puzzle is verified unique.

Intro copy

This lesson has one prerequisite that ordinary technique lessons do not:

The puzzle must have exactly one solution.

If you do not want to use uniqueness-based techniques, you can skip the lesson without losing the standard Sudoku rule set.

Step 1 — Make the premise explicit

Display puzzle metadata:

Solution count verified: YES
Unique solution: YES

Prompt:

Is Unique Rectangle reasoning allowed?

Yes.

Then show:

Unique solution: NOT VERIFIED

Prompt:

Can we safely use the technique?

No.

Feedback:

Uniqueness techniques need the extra one-solution premise.

Step 2 — Build the deadly geometry

Learner places four corner cells until they occupy:

  • 2 rows;
  • 2 columns;
  • exactly 2 boxes.

Then fill all with {4,9}.

Button:

Show possible completions

Display both diagonal swaps.

Prompt:

What is wrong with this state for a verified unique puzzle?

Correct:

It permits two valid completions.

Step 3 — Geometry near misses

Show:

  1. correct 2-row/2-column/2-box rectangle;
  2. four corners spanning four boxes;
  3. cells using three rows;
  4. correct geometry but one corner fixed in a way that breaks the simple swap.

Learner marks which one matches the standard UR setup.

Step 4 — Type 1 construction

Start:

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

Prompt:

What happens if D uses 2 or 7?

Show deadly rectangle.

Then:

Which candidates can be removed from D?

Correct:

  • 2;
  • 7.

Leave:

  • 9.

Step 5 — Guided verified puzzle

The UI must visibly confirm:

  • puzzle is unique;
  • example state is solver-validated.

Learner:

  1. identifies geometry;
  2. marks UR digits;
  3. identifies extra-candidate corner;
  4. performs Type 1 elimination.

Step 6 — Ordinary logic vs uniqueness

Present two states.

State A

A Hidden Single is available.

State B

Only the UR Type 1 elimination is being targeted.

Prompt:

Which move should you prefer in State A?

Hidden Single.

Feedback:

Uniqueness is optional advanced logic. Use a simpler ordinary deduction when it is available.

Practice

Every exercise must carry:

uniqueSolutionVerified = true

Practice pool:

  • valid Type 1;
  • four-box near miss;
  • extra-candidate wrong corner;
  • unverified-puzzle rejection;
  • simpler-deduction-first scenario.

Hint ladder

  1. “Check puzzle uniqueness.”
  2. “Look for four cells in two rows and two columns.”
  3. “Do they occupy exactly two boxes?”
  4. “Which two candidates define the potential deadly rectangle?”
  5. “Which corner contains the extra candidate?”
  6. Reveal Type 1 elimination.

Completion

  • state the uniqueness premise correctly;
  • classify deadly geometry;
  • solve three Type 1 examples;
  • reject one unverified puzzle;
  • choose simpler ordinary logic when available.

Next

Advanced Mixed Practice

Now decide between Fish, Wings, Chains, and optional uniqueness without the technique name being given first.

Reference

Read Unique Rectangle and Unique Sudoku Solutions for the full premise and product-validation requirements.