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Naked Quad

Learn how a Naked Quad reserves four digits across four cells in one Sudoku unit and how to recognize productive Quad eliminations without relying on identical candidate lists.

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A Naked Quad is a four-cell Naked Subset.

Inside one row, column, or box, choose four unsolved cells. If the union of every candidate in those four cells contains exactly four distinct digits, those four digits are reserved for those four cells.

The digits must occupy the four cells in some order.

So those same four digits can be removed from every other cell in the shared unit.

The core relationship is:

4 cells are restricted to 4 digits.

A Naked Quad is not a fundamentally new proof after Naked Pairs and Naked Triples. It is the same capacity argument at size 4.

What changes is recognition: four-cell combinations are easier to overlook, and the candidate lists rarely look identical.

Quick rule

A Naked Quad requires:

  1. four unsolved cells in one shared row, column, or box;
  2. exactly four distinct candidate digits across those four cells;
  3. no candidate outside that four-digit set in any of the four cells;
  4. at least one occurrence of a Quad digit elsewhere in the unit if the pattern is to make an immediate elimination.

Then:

remove the four Quad digits from every other cell in that unit.

The candidate-union test

The safest way to verify a Naked Quad is to merge the candidates from the four proposed cells.

Suppose four cells contain:

A = {1,4}
B = {1,7}
C = {4,9}
D = {7,9}

The combined set is:

{1,4,7,9}

There are:

  • 4 cells;
  • 4 distinct digits.

That is a valid Naked Quad.

Those cells must eventually contain 1, 4, 7, and 9 in some order. Therefore 1, 4, 7, and 9 cannot be used by any other cell in the same unit.

A Naked Quad does not require four identical candidate lists

The obvious-looking pattern:

{1,4,7,9}
{1,4,7,9}
{1,4,7,9}
{1,4,7,9}

is valid, but it is not the shape you should expect most often.

These can also form a Quad:

{1,4}
{1,7}
{4,9}
{7,9}

or:

{1,4,7}
{1,9}
{4,7,9}
{1,4}

or many other distributions.

The individual cells may contain two, three, or four Quad digits.

The only requirement is that the total union across the four cells contains exactly four digits.

If even one of the four cells contains an additional candidate outside that set, the proposed Naked Quad fails.

Why Naked Quads work

Four unsolved cells need four final values.

If all four cells are restricted to only four digits — call them W, X, Y, and Z — then those four digits must fill the four cells.

We do not need to know which cell receives which digit.

The set has exactly enough capacity for:

  • four cells;
  • four distinct final values.

There is no spare copy of W, X, Y, or Z available for another cell in the same row, column, or box.

This is the same reservation proof as:

Naked Pair   → 2 cells / 2 digits
Naked Triple → 3 cells / 3 digits
Naked Quad   → 4 cells / 4 digits

Learning that scalable rule is more useful than memorizing a special Quad picture.

Step-by-step board example

Naked Quads can appear in rows, columns, or boxes

The proof works in any house.

A Naked Quad can occupy four cells in:

  • one row;
  • one column;
  • one 3×3 box.

The four cells do not need to be adjacent.

They only need to share the complete unit in which you are making the eliminations.

Why there is no standard Locked Quad

Pairs and Triples can sometimes fit entirely inside the intersection between:

  • one row and one box;
  • or one column and one box.

That can let the subset eliminate candidates from two houses and is often called a Locked Pair or Locked Triple.

A row-box or column-box intersection contains at most three cells.

A four-cell subset therefore cannot fit entirely inside such an intersection.

That is why standard Sudoku terminology does not normally include a Locked Quad analogous to Locked Pairs and Locked Triples.

Naked Quad vs Naked Triple

The only structural change is size.

Naked Triple

3 cells → union of exactly 3 digits.

Naked Quad

4 cells → union of exactly 4 digits.

Quads are harder to spot because:

  • there are more combinations of cells to test;
  • candidate lists can be less visually similar;
  • simpler subsets may be hidden inside the same unit;
  • the pattern may already be nonproductive.

The proof itself is not harder.

Naked Quad vs Hidden Quad

Naked and Hidden Quads are dual views of the same four-by-four reservation.

Naked Quad

Start with four cells.

Ask:

Which digits can these cells contain?

If their candidate union is exactly four digits, remove those four digits from other cells in the unit.

Hidden Quad

Start with four digits.

Ask:

Where can these digits go in the unit?

If every possible position for those four digits is confined to the same four cells, remove unrelated candidates from those four cells.

After a Hidden Quad is cleaned up, those cells may visibly become a Naked Quad.

A valid Quad is not always a useful Quad

Suppose four cells contain only {1,4,7,9}, but no other cell in the unit contains 1, 4, 7, or 9.

The reservation is logically true.

But it removes nothing.

For solving purposes, distinguish:

  • a valid subset;
  • a productive subset.

Productive Quads create at least one candidate elimination.

This distinction becomes increasingly important as subsets grow larger because many theoretically valid sets do not advance the puzzle.

Watch for simpler subsets first

Before accepting a Naked Quad, check whether a smaller subset already explains the useful restriction.

For example, if two of the four cells form a Naked Pair and that Pair already creates the relevant eliminations, solving the Pair first is clearer and cheaper.

Likewise, a Naked Triple may be embedded in the same candidate pattern.

A good solving order is:

  1. Singles;
  2. Locked Candidates;
  3. Pairs;
  4. Triples;
  5. Quads.

Quads are useful, but they should not replace simpler deductions that are already available.

How to find Naked Quads efficiently

Trying every four-cell combination in every house is possible but inefficient for a human solver.

Use candidate structure instead.

1. Work one house at a time

Choose a constrained row, column, or box.

2. Focus on low-candidate cells

Cells with two or three candidates are especially useful starting points.

3. Merge candidate sets progressively

Do not search for four matching lists.

Instead ask whether four promising cells together stay inside a four-digit universe.

4. Check for smaller subsets

If two cells already form a Pair, or three form a Triple, solve the smaller pattern first when it is productive.

5. Verify every candidate in all four cells

One overlooked fifth digit invalidates the Naked Quad.

6. Look outside the Quad cells

The purpose of the pattern is to eliminate the Quad digits from other cells in the shared unit.

Common mistakes

Requiring identical candidate lists

Not required. The union matters.

Counting four cells but five digits

Four cells restricted to five distinct digits are not a Naked Quad.

Ignoring an extra candidate in one Quad cell

A candidate outside the proposed four-digit set breaks the reservation.

Removing candidates from the Quad cells

The Quad digits must remain possible inside the four cells.

Remove them from other cells in the house.

Using cells that do not share one complete unit

All four cells must belong to the same row, column, or box for the Naked Subset elimination.

Overlooking a simpler Pair or Triple

Solve cheaper logic first.

Treating a nonproductive Quad as progress

A valid pattern with no eliminations does not change the candidate state.

Recognition drill

Inside one row, suppose four cells contain:

{2,6}
{2,8}
{6,9}
{8,9}

Ask:

  1. How many cells are selected?
  2. What is the union of their candidates?
  3. How many distinct digits are in that union?
  4. Do any other cells in the row contain 2, 6, 8, or 9?

The answers are:

  • four cells;
  • {2,6,8,9};
  • four digits;
  • if yes, those outside occurrences are elimination targets.

FAQ

Does every Naked Quad cell need four candidates?

No. Individual cells may contain two, three, or four of the Quad digits.

Can four bivalue cells form a Naked Quad?

Yes, if their combined candidate union contains exactly four distinct digits.

Can a Naked Quad appear in a box?

Yes. Naked Subsets work in rows, columns, and boxes.

Is a Naked Quad just a larger Naked Triple?

Yes. It uses the same N cells → N digits reservation rule with N = 4.

Is there a Locked Quad?

Not in the standard row/column-box sense. The intersection of a line and a box contains only three cells, so four Quad cells cannot all occupy that intersection.

Are Naked Quads common?

They are less common and harder to recognize than Pairs or Triples, but they are a standard subset technique and can produce useful eliminations in candidate-heavy puzzles.

Should I search for Quads before Fish or Wings?

Usually search simpler subsets first. After candidate cleanup, the next useful technique depends on the puzzle structure rather than a universal fixed order.

What to learn next

Compare this cell-first view with Hidden Quad, then return to the Fish family with X-Wing, Swordfish, and Jellyfish to see the same constrained-capacity idea applied across multiple houses for one digit.

Techniques · IntermediateNaked TripleLearn the Naked Triple rule: three cells whose combined candidates contain exactly three digits, allowing eliminations elsewhere in the unit.Techniques · AdvancedHidden QuadLearn the Hidden Quad rule in Sudoku: four digits confined to four cells, how to verify their position union, and which candidates can be removed.Techniques · IntermediateHidden TripleLearn how three digits restricted to the same three cells form a Hidden Triple and which distracting candidates can be removed.Techniques · AdvancedX-WingLearn the X-Wing Sudoku technique, how two rows and two columns form a size-2 Fish, and where the resulting candidate eliminations are valid.