Naked Quads in Sudoku: The Complete Guide

Naked Quads in Sudoku: The Complete Guide

What is a Naked Quad?

A naked quad (also called a "naked quadruple") occurs when four cells in the same unit (row, column, or box) collectively contain only four candidate numbers between them. These four numbers are "locked" into those four cells, which means you can eliminate them from all other cells in that unit.

Like naked pairs and naked triples, the cells don't need identical candidates—they just need to use exactly four distinct numbers between them.

Example:

In Box 3 (top-right), you have four cells:

  • Cell A: {2, 5, 8}
  • Cell B: {2, 5}
  • Cell C: {5, 8, 9}
  • Cell D: {2, 9}

Together, these four cells use only four distinct candidates: 2, 5, 8, and 9. This is a naked quad. Since these numbers must go in these four cells (in some arrangement), you can eliminate 2, 5, 8, and 9 from all other cells in Box 3.

Why Naked Quads Matter

  • They solve advanced puzzles where simpler techniques fail.
  • They can eliminate candidates from multiple cells simultaneously.
  • They're less common than pairs and triples, making them satisfying to find.
  • They complete the "naked subset" family (singles, pairs, triples, quads).
  • They demonstrate mastery of subset elimination logic.

How to Spot a Naked Quad

  1. Look for four cells in the same unit.
    Scan one row, column, or box at a time, focusing on cells with few candidates.
  2. Count the distinct candidates.
    Across the four cells, count how many unique numbers appear. If exactly four, you may have a quad.
  3. Verify the pattern.
    Common patterns include various distributions like {A,B}, {C,D}, {A,C}, {B,D} or {A,B,C,D} appearing across the four cells in different combinations.
  4. Eliminate from other cells.
    Remove all four candidates from every other cell in that unit.

Naked Quad vs Other Naked Subsets

The naked subset family all follows the same principle:

  • Naked pair: 2 cells with 2 candidates → eliminate from other cells
  • Naked triple: 3 cells with 3 candidates → eliminate from other cells
  • Naked quad: 4 cells with 4 candidates → eliminate from other cells

All eliminate outward from the group to other cells. Naked quads are simply the largest practical subset humans typically search for.

Step-by-Step Example

Let's examine Row 5:

Row 5 candidates:

  • Col 1: {1, 3, 6, 7}
  • Col 2: {1, 3, 7}
  • Col 3: [filled with 4]
  • Col 4: {2, 5, 8, 9}
  • Col 5: {1, 3, 6}
  • Col 6: {2, 5, 8, 9}
  • Col 7: {2, 5, 9}
  • Col 8: {1, 6, 7}
  • Col 9: {2, 8, 9}

Analysis: Look at columns 1, 2, 5, and 8:

  • Col 1: {1, 3, 6, 7}
  • Col 2: {1, 3, 7}
  • Col 5: {1, 3, 6}
  • Col 8: {1, 6, 7}

Count distinct candidates: 1, 3, 6, 7—exactly four numbers!

Naked quad found: {1, 3, 6, 7} across these four cells in Row 5.

Elimination: Remove 1, 3, 6, and 7 from all other cells in Row 5. In this case:

  • Columns 4, 6, 7, and 9 don't have any of these numbers, so no eliminations in this example
  • But in a real puzzle, other cells might contain these candidates and would be cleaned up

Visual Example: The Perfect Quad

  • Scenario: In Column 7, you find four cells with:
  • Cell A: {2, 4, 6, 8}
  • Cell B: {2, 4, 6, 8}
  • Cell C: {2, 4, 6, 8}
  • Cell D: {2, 4, 6, 8}
  • Pattern: All four cells have identical candidates—the clearest possible naked quad.
  • Naked quad confirmed: {2, 4, 6, 8} are locked to these four cells.
  • Elimination: Remove 2, 4, 6, and 8 from all other cells in Column 7.
  • Result: The remaining cells become much simpler, often revealing naked singles.

Strategies for Spotting Naked Quads Quickly

  1. Focus on cells with 2-3 candidates
    While quads can involve cells with 4 candidates, most practical examples have smaller subsets within the four cells.
  2. Look for repeating patterns
    If you see the same 3-4 numbers appearing repeatedly in a unit, investigate for a quad.
  3. Check after finding triples
    Sometimes what looks like a triple is actually part of a larger quad.
  4. Systematic unit scanning
    Go through each row, column, and box methodically rather than jumping around.
  5. Don't overinvest time
    Naked quads are rare. Don't spend too long searching—they're more of a "nice bonus" than a go-to technique.

Common Pitfalls

  • Counting wrong: Make sure there are exactly four distinct candidates across the four cells, not three or five.
  • Looking across multiple units: All four cells must be in the same row, column, OR box—not spread across different units.
  • Missing non-identical patterns: The cells don't all need {1,2,3,4}. Various combinations work as long as only four numbers appear total.
  • Forgetting to eliminate: After finding a quad, actually remove those candidates from other cells.
  • Searching too hard: Naked quads are rare in practice. If you can't find one quickly, move on to other techniques.

Practice: Find the Naked Quad

Try this Box 8 (bottom-middle) scenario:

Box 8 cells (rows 7-9, columns 4-6):

  • Row 7, Col 4: {3, 5, 7, 9}
  • Row 7, Col 5: {3, 5, 9}
  • Row 7, Col 6: [filled with 1]
  • Row 8, Col 4: {2, 4, 6, 8}
  • Row 8, Col 5: {5, 7, 9}
  • Row 8, Col 6: {2, 4, 6, 8}
  • Row 9, Col 4: {3, 5, 7}
  • Row 9, Col 5: {2, 4, 6, 8}
  • Row 9, Col 6: {3, 5, 9}

Question: Can you find a naked quad?

Solution: Look at the candidates containing numbers from the 2, 4, 6, 8 set:

  • Row 8, Col 4: {2, 4, 6, 8}
  • Row 8, Col 6: {2, 4, 6, 8}
  • Row 9, Col 5: {2, 4, 6, 8}

Wait, that's only three cells. Let me reconsider...

Looking at 3, 5, 7, 9 instead:

  • Row 7, Col 4: {3, 5, 7, 9}
  • Row 7, Col 5: {3, 5, 9}
  • Row 8, Col 5: {5, 7, 9}
  • Row 9, Col 4: {3, 5, 7}
  • Row 9, Col 6: {3, 5, 9}

That's five cells—too many. Let me provide a clearer example:

Clearer Practice Example - Column 2:

Column 2 candidates:

  • Row 1: {1, 4}
  • Row 2: [filled]
  • Row 3: {1, 4, 7, 9}
  • Row 4: {2, 3, 5, 6, 8}
  • Row 5: {1, 7, 9}
  • Row 6: {2, 3, 5, 6, 8}
  • Row 7: {4, 7, 9}
  • Row 8: {2, 3, 5, 6, 8}
  • Row 9: [filled]

Check: Rows 1, 3, 5, and 7 collectively use which numbers?

  • Row 1: {1, 4}
  • Row 3: {1, 4, 7, 9}
  • Row 5: {1, 7, 9}
  • Row 7: {4, 7, 9}

Distinct candidates: 1, 4, 7, 9—exactly four!

Naked quad found: {1, 4, 7, 9}

Action: Eliminate 1, 4, 7, and 9 from rows 4, 6, and 8 in Column 2. Since those rows contain {2, 3, 5, 6, 8}, there are no eliminations in this specific example, but the quad is confirmed.

Why Naked Quads Set the Stage

Naked quads represent the pinnacle of subset elimination logic:

  • They complete your understanding of naked subsets (singles → pairs → triples → quads)
  • They demonstrate that larger patterns follow the same logic as smaller ones
  • They prepare you for advanced techniques that involve tracking multiple constraints
  • They're satisfying to find and usually indicate you're working on a challenging puzzle
  • They bridge intermediate techniques with expert-level pattern recognition

Quick Recap

Technique Cells Candidates Rarity Difficulty
Naked Pair 2 2 Common Intermediate
Naked Triple 3 3 Occasional Intermediate
Naked Quad 4 4 Rare Advanced
Hidden Quad 4 4 Very Rare Advanced

Final Thought

Naked quads are the "unicorns" of Sudoku—rare but powerful. When you find one, you know you're solving a truly challenging puzzle. Are four cells sharing just four candidates? You might have found a naked quad!

Frequently Asked Questions

What is a Naked Quad in Sudoku?

A naked quad occurs when four cells in the same unit (row, column, or box) collectively contain only four candidate numbers between them. These four numbers are locked into those four cells, allowing you to eliminate them from all other cells in that unit. Each cell may have 2-4 candidates, but together they use exactly four distinct numbers.

How do I spot a Naked Quad?

To spot naked quads: 1) Look for four cells in the same unit with limited candidates, 2) Check if they collectively contain only four distinct candidate numbers, 3) Verify the pattern (various combinations like {1,2}, {3,4}, {1,3}, {2,4} or {1,2,3,4} repeated), 4) Eliminate those four numbers from all other cells in the unit.

What's the difference between Naked Quads and Naked Triples?

Naked triples involve three cells with three candidates; naked quads involve four cells with four candidates. Both eliminate outward from the group. Naked quads are rarer than triples but can eliminate from more cells. The logic is identical—just scaled from 3 cells to 4 cells.

Why are Naked Quads important?

Naked quads are important because they solve advanced puzzles where simpler techniques fail, they can eliminate candidates from multiple cells simultaneously, they're less common but highly effective when found, and they complete the naked subset family (singles, pairs, triples, quads).

When should I look for Naked Quads?

Look for naked quads after exhausting naked singles, pairs, and triples. They're rare but appear in difficult puzzles. Systematically scan units where many cells have 2-4 candidates each, looking for four cells that share only four distinct candidates between them.

Ready to advance? Check out our complete strategy guide for more techniques.

Related Strategies

Once you've mastered naked quads, these techniques build naturally:

  • Naked Triples - The foundation for understanding quads
  • Naked Pairs - The simplest naked subset technique
  • Hidden Quads - The complementary technique (coming soon)
  • X-Wing - Advanced pattern recognition
  • XY-Wing - Advanced three-cell pattern

Practice Naked Quads

Next up: Try Hidden Quads to complete your quad techniques toolkit (coming soon).

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