Rectangle Elimination

Rectangle Elimination is a chaining technique that uses a rectangular pattern of strong and weak links to find eliminations. It combines the logic of conjugate pairs with weak link chains to prove that certain candidates are impossible.

How It Works

The Pattern

The technique requires four cells forming a rectangle across four boxes:

  1. Hinge cell — Has a strong link (conjugate pair) with another cell
  2. Strong link partner — The other end of the conjugate pair (different box)
  3. Wing 1 — Has a weak link to the hinge (perpendicular direction)
  4. Wing 2 — Completes the rectangle (same row/column as both partners)

The Logic Chain

The elimination works through a chain of implications:

  1. Assume Wing 1 contains the digit
  2. Then Hinge cannot contain the digit (weak link — same row or column)
  3. Then Strong partner MUST contain the digit (conjugate pair — if one is OFF, other is ON)
  4. Then Wing 2 cannot contain the digit (weak link — same row or column as strong partner)
  5. Check the fourth corner box — if ALL its candidates for this digit would be eliminated, we have a contradiction
  6. Therefore Wing 1 cannot contain the digit — eliminate it!

Why It Works

The key insight is that a conjugate pair (strong link) creates a forcing chain. If one cell in the pair is OFF, the other MUST be ON. By building a rectangle around this pair, we can prove that certain positions lead to impossible situations.

Example

S9B027y070n828y8r081m05010804028i8m071i7y067q0n0807057v027y027q060108077y05088e060716027y7q01071601828a030206087q0508067u0u0207060704020301080509q080282078a1r0n1q

In this puzzle:

The logic chain:

  1. If R1C2 = 9
  2. Then R7C2 ≠ 9 (same column)
  3. Then R7C6 = 9 (conjugate pair forces it)
  4. Then R1C6 ≠ 9 (same column)
  5. Box 2 (containing R1C6) would lose all 9 candidates — contradiction!
  6. Therefore: R1C2 ≠ 9 — eliminate 9 from R1C2

Visual Pattern

      C2          C6
       ┌──────────┐
R1  Wing1 ─ ─ ─ Wing2
       │weak     │
       │         │
R7  Hinge ═════ Strong
       strong link

Row vs Column Based

Row-based: Strong link is in the same row; weak links are in columns Column-based: Strong link is in the same column; weak links are in rows

Both work identically — the rectangle just rotates 90°.

Complexity

Rectangle Elimination is an advanced technique because it requires:

The app's hint system handles this automatically, highlighting all four cells and the elimination.

Tips

  1. Look for conjugate pairs first — These are the foundation of the pattern
  2. Span four boxes — The pattern must cross box boundaries to work
  3. Check the fourth box — The contradiction must eliminate ALL candidates there
  4. Use the chain visualiser — Strong links show potential starting points

More Puzzles

Related Techniques