Hidden Quad

A Hidden Quad is a four-digit Hidden Subset.

Inside one row, column, or box, choose four candidate digits. If every legal position for those four digits is confined to the same four cells, then those four cells are reserved for the four digits.

The cells may still display many unrelated candidates.

Those extra candidates can be removed.

The core relationship is:

4 digits are confined to 4 cells.

Hidden Quads use exactly the same positional logic as Hidden Pairs and Hidden Triples, scaled to four digits.

They are called hidden because the important reservation can be buried underneath many other pencil marks.

Quick rule

A Hidden Quad requires:

1. one shared row, column, or box;
2. four candidate digits;
3. every occurrence of those four digits in that unit confined to the same four cells;
4. at least one unrelated candidate inside those cells if the Quad is to create an immediate elimination.

Then:

remove every non-Quad candidate from those four cells.

The position-union test

Hidden Subsets are easiest to verify by tracking where digits can go, not by comparing complete cell candidate lists.

Suppose in one column:

2 → A/C
4 → A/B/D
7 → B/C
9 → C/D

The union of all possible positions for 2, 4, 7, and 9 is:

{A,B,C,D}

There are:

• 4 digits;
• 4 cells containing every possible position for those digits.

That is a Hidden Quad.

The four cells must eventually receive 2, 4, 7, and 9 in some order.

Any other candidates written in A, B, C, or D can therefore be removed.

Hidden Quad cells can contain many extra candidates

A Hidden Quad may be visually noisy.

For example:

A = {1,2,4,6}
B = {3,4,7,8}
C = {2,5,7,9}
D = {1,4,6,9}

If, within the source unit, the digits 2, 4, 7, and 9 occur nowhere outside A/B/C/D, then those four digits are reserved for those four cells.

The unrelated candidates can be removed:

A → {2,4}
B → {4,7}
C → {2,7,9}
D → {4,9}

The hidden reservation has become visible.

This is why full, trustworthy candidate notation matters so much for Hidden Subsets.

Why Hidden Quads work

A row, column, or box must eventually contain each required digit once.

If four digits can appear only in four cells of that unit, then those four cells must collectively provide the placements for all four digits.

No fifth value can occupy one of those four cells without taking away a required position from the reserved set.

Therefore every unrelated candidate in those four cells is impossible.

The scalable Hidden Subset rule is:

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

Step-by-step board example

Hidden Quads can appear in rows, columns, or boxes

The source house may be:

• a row;
• a column;
• a 3×3 box.

The four cells do not need to be adjacent.

What matters is that the four target digits have no legal position elsewhere in that same unit.

Candidate occurrences in other houses are irrelevant to the proof unless they also lie in the source unit.

Hidden Quad vs Hidden Triple

The technique scales directly.

Hidden Triple

3 digits → all positions confined to 3 cells.

Hidden Quad

4 digits → all positions confined to 4 cells.

Quads are harder to recognize because:

• more digits must be tracked at once;
• each digit may appear in two, three, or four of the target cells;
• the cells may contain many distracting candidates;
• smaller Hidden Subsets should usually be solved first.

Hidden Quad vs Naked Quad

The two descriptions are dual.

Hidden Quad

Start from digit positions.

If four digits are confined to four cells, remove other candidates from those cells.

Naked Quad

Start from cell contents.

If four cells contain only four digits in total, remove those digits from other cells in the unit.

After a Hidden Quad is cleaned, the same four cells may form a visible Naked Quad.

That does not mean one technique was wrong. They are two ways of describing the same underlying reservation.

Hidden Quad vs four low-frequency digits

Four digits each appearing only a few times are not automatically a Hidden Quad.

The decisive test is the position union.

For example:

2 → A/B
4 → B/C
7 → C/D
9 → D/E

Each digit has only two positions, but together they occupy five cells:

{A,B,C,D,E}

That is not a Hidden Quad.

You need four digits whose complete position union is confined to exactly four cells.

A target digit with one position is a Hidden Single

If one of the proposed Quad digits appears in only one cell of the unit, solve the Hidden Single first.

For example:

2 → A
4 → A/B/D
7 → B/C
9 → C/D

Candidate 2 is already forced into A.

After placing it and updating candidates, the remaining structure may simplify into a Pair or Triple.

Prefer the simpler deduction.

A valid Hidden Quad is not always productive

Suppose four digits are confined to four cells, but those cells already contain only the four target digits.

The Hidden Quad is logically present.

But there are no unrelated candidates to remove.

That makes it nonproductive at the current state.

Hidden Subsets are most useful when they clean significant candidate clutter from the reserved cells and reveal follow-up deductions.

How to find Hidden Quads efficiently

Hidden Quads are difficult to spot by staring at complete candidate lists.

Search digit-first.

1. Choose one unit

Work inside one row, column, or box.

2. Identify low-frequency digits

Digits appearing in two, three, or four unsolved cells are natural candidates.

3. Combine four candidate-position maps

Track the union of the cells in which the four digits can appear.

4. Look for a four-cell union

If all four digits are confined to exactly four cells, verify the Quad.

5. Check for smaller subsets first

Two digits confined to two cells form a Hidden Pair.

Three digits confined to three cells form a Hidden Triple.

Solve those first when available.

6. Remove only unrelated candidates

The four target digits must remain in the four reserved cells.

Common mistakes

Looking for four similar cells

Hidden Quads are defined by digit positions, not by matching candidate lists.

Missing one occurrence elsewhere in the source unit

If one target digit is legal in a fifth cell, the four-cell confinement fails.

Removing the target digits

Keep the four reserved digits. Remove unrelated candidates from the four cells.

Using incomplete pencil marks

A missing candidate occurrence can create a false Hidden Quad. Use trustworthy candidate data.

Treating a Hidden Single as part of a Quad

If one digit has one legal position, solve the Single first.

Ignoring a simpler Hidden Pair or Triple

Smaller subsets are easier to verify and usually preferred.

Recognition drill

Inside one box, suppose:

1 → A/B
3 → A/C/D
6 → B/C
8 → C/D

Ask:

1. How many digits are being tracked?
2. What is the union of all their possible cells?
3. Are any of the four digits possible elsewhere in the box?
4. Do A/B/C/D contain candidates other than 1, 3, 6, and 8?

If the position union is exactly four cells and no target digit appears outside them, you have a Hidden Quad. Any unrelated candidates inside the four cells can be removed.

FAQ

Does each Hidden Quad digit need four possible positions?

No. Each target digit may occupy two, three, or four of the four reserved cells.

Can a Hidden Quad cell contain many candidates?

Yes. Extra candidates are what make the subset hidden, and they are the candidates the technique removes.

Can a Hidden Quad appear in a box?

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

Can a Hidden Quad become a Naked Quad?

Yes. After unrelated candidates are removed, the four reserved cells contain only the four Quad digits.

Are Hidden Quads common?

They are relatively uncommon and harder to recognize than smaller subsets, especially without complete pencil marks.

Is a Hidden Quad more advanced than a Naked Quad?

Both use the same four-by-four reservation. Hidden Quads are often harder for humans to spot because the important digits are buried inside larger candidate lists.

What to learn next

Compare the digit-first Hidden Quad with Naked Quad, then move into Jellyfish to see another size-4 constrained-set pattern operating across rows and columns for a single candidate digit.