Naked Quads

Naked quads are the natural extension of naked pairs and triples. Four cells, four candidates, same elimination logic. They're rarer and harder to spot, but powerful when you find them.

What is a Naked Quad?

A naked quad occurs when four cells in the same unit contain only candidates from a set of four numbers. These four cells claim those four digits completely.

The pattern:

• 4 cells
• 4 candidates distributed among them
• No other candidates in those cells
• All four digits must appear somewhere in the four cells

Why Quads Are Harder to See

With pairs, you're matching two cells. Easy.

With triples, three cells. Manageable.

With quads, four cells with up to four candidates each means many possible combinations. Your brain has to:

• Track more cells
• Compare more candidates
• Recognize less obvious patterns

This is why quads often go unnoticed even by experienced solvers.

Valid Quad Combinations

For digits 1, 3, 5, 7:

Key requirement: Each of the four digits must appear in at least one of the four cells.

The Logic

If four cells contain only 1, 3, 5, 7:

• These cells will take all four digits (one per cell)
• The exact distribution is unknown
• But those four digits are definitely claimed

Result: No other cell in the unit can contain 1, 3, 5, or 7.

Step-by-Step Example

Before elimination:

Finding the quad:

Look for cells with limited, overlapping candidates:

• Cell B: [3,5,7]
• Cell C: [1,3]
• Cell E: [1,5,7]
• Cell F: [1,3,7]

Collective candidates: 1, 3, 5, 7 — exactly four digits in four cells.

Verification:

• Does 1 appear? Yes (C, E, F)
• Does 3 appear? Yes (B, C, F)
• Does 5 appear? Yes (B, E)
• Does 7 appear? Yes (B, E, F)

This is a valid naked quad!

After elimination:

Check for new patterns or singles in the simplified grid.

How to Find Naked Quads

Method 1: Build Up from Pairs

1. Find a naked pair
2. Look for two more cells with subsets of related digits
3. If four cells share exactly four digits, you have a quad

Method 2: Candidate Census

1. For each unit, list which cells contain which candidates
2. Group candidates that frequently appear together
3. If four candidates appear only in four cells, that's a quad

Method 3: Elimination Thinking

1. Notice a unit with many eliminations needed
2. Ask: "What if these four cells had all of digits X, Y, Z, W?"
3. Verify the pattern holds

Method 4: Process of Elimination

Sometimes it's easier to find what ISN'T the quad:

1. A unit has 6 empty cells
2. Two cells clearly have candidates outside your suspected quad
3. The remaining four cells might be your quad

Quads in Boxes

Box quads are particularly useful:

┌─────────────────────┐
│ [1,3,5,7] │ [2,8,9] │ [1,5]   │
│ [3,7]     │ [4,6]   │ [2,4,9] │
│ [1,5,7]   │ [2,6,9] │ [3,5]   │
└─────────────────────────────────┘

Cells with [1,3,5,7], [1,5], [3,7], [3,5] collectively contain only 1, 3, 5, 7... let's check:

• [1,3,5,7]: all four
• [1,5]: 1, 5
• [3,7]: 3, 7
• [1,5,7]: 1, 5, 7

That's FIVE cells. Not a quad.

Let's try: [1,5], [3,7], [1,5,7], [3,5]:

• Digits covered: 1, 3, 5, 7
• Four cells: ✓
• Only these four digits: ✓

Naked quad confirmed! Eliminate 1, 3, 5, 7 from [1,3,5,7] → becomes empty (error check needed in real solving).

The Quad-Pair Connection

Here's a useful insight: in a unit with exactly 6 empty cells:

• If there's a naked quad, the other 2 cells form a naked pair on the remaining digits
• If there's a naked pair, the other 4 cells might form a naked quad

Example:

Unit with 6 empty cells, candidates [1,3], [3,5], [5,7], [1,7], [2,9], [2,9]:

• First four cells: quad on 1,3,5,7
• Last two cells: pair on 2,9

Finding either pattern reveals the other!

Naked Quads vs. Hidden Quads

When you find a naked quad, you eliminate those digits from OTHER cells.

When you find a hidden quad, you eliminate OTHER digits from THESE cells.

Practical Tips

Don't Hunt Blindly

Quads are rare. Don't waste time searching if simpler techniques work.

Look for quads when:

• Basic techniques are exhausted
• A unit has 5-6 empty cells with overlapping candidates
• You see near-pairs that don't quite work

Use Paper (or Mental Marking)

For quads, it helps to:

• Jot down cell positions and their candidates
• Look for digit groupings
• Physically verify the pattern

Trust the Math

If four cells genuinely share only four candidates:

• The quad is valid
• Eliminations are safe
• Don't second-guess

Quad Frequency

In typical puzzles:

• Easy/Medium: Almost never needed
• Hard: Occasionally
• Expert: Sometimes
• Evil: More common

Practice Exercise

Find the naked quad:

Row candidates:

[2,4,6] | [1,3,6] | [3,4,5] | [1,5,6] | [1,3,4,5] | [2,4] | [1,5] | [2,6] | (filled)

Which four cells form the quad? What digits? What eliminations?

Common Mistakes

Mistake 1: Miscounting cells or digits

Always verify:

• Exactly four cells
• Exactly four digits total
• Each digit appears at least once

Mistake 2: Including cells with extra candidates

A cell with [1,3,5,7,9] cannot be part of a [1,3,5,7] quad.

Mistake 3: Confusing with hidden quads

Remember: naked = these cells have ONLY these candidates.

Mistake 4: Missing one cell of the quad

If you find three cells with three overlapping candidates, look for a fourth cell that might complete a quad.

Quick Reference

Naked quad formula:

• 4 cells + 4 candidates = naked quad
• No other candidates in those cells
• All four digits must be represented

Finding quads:

1. Look in units with 5-6 empty cells
2. Find cells with 2-4 candidates
3. Check if four cells share exactly four digits
4. Verify no extra candidates

Applying quads:

1. Eliminate the four digits from all OTHER cells in the unit
2. Check for resulting singles
3. Look for new patterns

When to hunt quads:

• Basic and intermediate techniques exhausted
• Unit has many overlapping candidates
• Hard/Expert+ puzzles