Sudoku Subsets

A Sudoku subset is a reservation between a set of cells and a set of digits inside one house.

The general idea is:

N cells are reserved for N digits.

Once that relationship is proved, the same digits cannot be used elsewhere in the house, or other candidates cannot remain inside the reserved cells, depending on whether the subset is Naked or Hidden.

Pairs, Triples, and Quads are not six unrelated tricks. They are different sizes and viewpoints of the same counting principle.

That makes subsets one of the most useful families to understand conceptually rather than memorize by shape.

Quick rule

A subset always lives inside a single house — one row, one column, or one 3×3 box.

For size N:

• Naked subset: N cells contain candidates drawn from only N digits. Those digits are reserved for those cells, so remove them from the other cells in the house.
• Hidden subset: N digits can appear only in N cells. Those cells are reserved for those digits, so remove other candidates from the subset cells.

The sizes are:

Why Sudoku subsets work

Suppose three cells in one row can contain only digits {2,5,8} between them.

You may not know which cell gets which digit, but the row still needs all three digits somewhere. Because those three cells are the only places available inside the subset, they must consume the three digits in some order.

Therefore the rest of the row cannot use 2, 5, or 8.

The proof does not depend on the individual arrangement being known.

It depends on a one-to-one reservation:

3 cells ↔ 3 digits

The Hidden version expresses the same reservation from the opposite direction.

If digits 2, 5, and 8 can occur only in three cells of a row, those three cells must contain those three digits. Any other candidates written in the three cells are impossible.

Naked subsets: start from the cells

A Naked Subset is usually found cell-first.

Look for a small group of unsolved cells whose combined candidate union contains no more digits than cells.

For a Naked Pair:

{3,7} {3,7}

Two cells contain exactly two digits in total.

For a Naked Triple, the cells do not need identical candidate lists:

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

The union is still only {1,4,7}.

For a Naked Quad, four cells may distribute four digits unevenly while their total union remains exactly those four digits.

The correct question is not:

Do these cells look the same?

It is:

How many distinct candidates are available across these N cells?

Hidden subsets: start from the digits

A Hidden Subset is usually found digit-first.

Instead of looking for small candidate lists, ask where particular digits can still appear inside the house.

Suppose digits 1, 4, and 7 occur nowhere in a row except three cells.

Those three digits must occupy those three cells in some order.

Even if the cells currently look like:

{1,3,4}
{1,4,7,9}
{2,4,7}

the candidates 2, 3, and 9 can be removed from the three cells because the cells are reserved for {1,4,7}.

That is why Hidden Subsets can be harder to see: the relevant cells may look noisy before the elimination.

Naked and Hidden subsets are two views of one reservation

Naked and Hidden Subsets are dual perspectives.

A Naked Subset emphasizes:

these cells have too few possible digits.

A Hidden Subset emphasizes:

these digits have too few possible cells.

In both cases, N cells become reserved for N digits.

This is useful strategically because it tells you how to switch search modes:

• if several cells have small candidate lists, search Naked;
• if several digits have restricted positions, search Hidden.

A worked subset example

Naked Pair

A Naked Pair is the smallest and usually easiest subset to recognize.

Two cells in the same house are restricted to the same two digits.

Those digits belong to the Pair, so remove them from every other unsolved cell in that house.

This is often the first subset technique worth learning because the reservation is visually obvious.

Use the dedicated Naked Pair Guide for examples, targets, common false patterns, and recognition drills.

Hidden Pair

A Hidden Pair reverses the search.

Two digits can occur in only two cells of the house. Those two cells may contain additional candidates, but the extra candidates can be removed.

Hidden Pairs are often easier to find by making a digit-position map mentally or from full candidate notation.

Naked and Hidden Triples

Triples extend the same logic from two to three.

The common mistake is requiring three identical three-candidate cells.

That is unnecessary.

For Naked Triples, three cells only need their combined candidate union to be three digits.

For Hidden Triples, three digits only need their combined possible positions to be three cells.

The pattern becomes less visually obvious as N grows, which is why understanding the reservation proof matters more than memorizing examples.

Naked and Hidden Quads

Quads use four cells and four digits.

They are logically no different from Pairs or Triples, but they are usually harder to spot and may produce fewer useful eliminations because so much of the house is already involved in the subset.

A Quad is worth searching for when:

• the candidate state is dense but accurate;
• Pairs and Triples have been exhausted;
• four cells share a restricted candidate union;
• or four digits are unusually restricted in one house.

How to find subsets efficiently

Do not scan every possible combination blindly.

Use structure to reduce the search.

1. Start with constrained houses

Rows, columns, or boxes with fewer unsolved cells are easier to analyze.

2. For Naked Subsets, start with small candidate cells

Bivalue and trivalue cells are natural anchors.

3. For Hidden Subsets, track restricted digits

Digits appearing in only two, three, or four positions inside the house are the natural candidates.

4. Stop when the count no longer closes

If three selected cells contain five distinct digits, they are not a Naked Triple.

If three selected digits can occur in five different cells, they are not a Hidden Triple.

5. Re-scan after every elimination

A subset is often only a setup move. Its eliminations may immediately create a Single or Locked Candidate.

The exact subset test

For a Naked Subset of size N:

1. choose N unsolved cells in one house;
2. take the union of their candidates;
3. the union must contain exactly N digits;
4. those digits are reserved for those N cells;
5. eliminate them from other cells in the house.

For a Hidden Subset of size N:

1. choose N digits in one house;
2. take the union of all cells where those digits can appear;
3. that union must contain exactly N cells;
4. those cells are reserved for those N digits;
5. eliminate other candidates from the reserved cells.

A subset does not require solved order

Subsets are reservation proofs, not placement proofs.

A Naked Pair {2,9} {2,9} does not tell you which cell is 2 and which is 9.

It tells you that those two cells consume 2 and 9 between them.

That is enough to eliminate the digits elsewhere.

The same applies to Triples and Quads.

Subsets can exist in rows, columns, and boxes

Every subset proof is local to one house.

Do not transfer an elimination into another house just because one of the subset cells happens to belong to it.

If a Naked Pair is proved in row 4, its Pair elimination applies to row 4.

A separate box elimination is allowed only if the same reservation is also valid inside that box.

This house-local rule prevents many false eliminations.

Locked subsets and overlap with intersections

Some subset terminology uses names such as Locked Pair or Locked Triple when a Naked Subset is confined in a way that also interacts with a box-line intersection.

For VeyraPlay, the important conceptual split is:

• use Subsets for the N-cells / N-digits reservation;
• use Locked Candidates for candidate confinement across a box and line.

A board may support both descriptions, but you should always be able to state the exact house and exact reservation that proves the elimination.

Common subset mistakes

Requiring identical candidate lists

Only Naked Pairs commonly appear as two identical bivalue cells.

Triples and Quads can use different candidate combinations as long as the union has the correct size.

Counting cells from different houses

A subset proof belongs to one row, column, or box.

Eliminating inside a Naked Subset

Naked Subsets eliminate the reserved digits from other cells in the house.

Eliminating outside a Hidden Subset

Hidden Subsets eliminate other candidates from the reserved cells.

Trusting incomplete candidates

A missing legal candidate can create a false subset.

Verify the candidate state before making an advanced elimination.

Searching Quads before obvious Pairs

Use the smallest useful logic first. Larger subsets are harder to recognize and often unnecessary.

Subsets vs Locked Candidates

Both can eliminate candidates without immediately solving a cell.

Locked Candidates use a box-line positional restriction for one digit.

Subsets reserve N digits across N cells inside one house.

The overlap in some diagrams does not make the proofs interchangeable.

Subsets vs Fish

Subsets work inside one house and involve several digits or several cells.

Fish follow one digit across multiple rows and columns using base and cover sets.

The counting flavor is related, but the objects being counted are different.

When should you learn Quads?

Learn the family in this order:

1. Naked Pair;
2. Hidden Pair;
3. Naked Triple;
4. Hidden Triple;
5. Naked Quad;
6. Hidden Quad.

Do not wait until every earlier pattern is instant before reading the next one, but make sure you understand the N-for-N reservation before increasing N.

FAQ

What is a subset in Sudoku?

A subset is an N-for-N reservation inside one house: N cells are reserved for N digits. That reservation allows candidate eliminations either outside the cells for Naked Subsets or inside the cells for Hidden Subsets.

What is the difference between a Naked and Hidden Subset?

Naked Subsets are usually found from restricted cell candidate lists. Hidden Subsets are usually found from digits that have restricted possible positions. They express the same underlying reservation from opposite directions.

Are Pairs, Triples, and Quads all subsets?

Yes. Pair means size 2, Triple size 3, and Quad size 4. Each size can be Naked or Hidden.

Can a Naked Triple contain a cell with only two candidates?

Yes. The three cells do not need three candidates each. Their combined candidate union only needs to contain exactly three digits.

Do subsets solve cells directly?

Usually no. They normally eliminate candidates. Those eliminations may then create Singles or other easier deductions.

Can subsets use cells from different rows or boxes?

The selected subset cells must share one house. A pattern made from cells that do not all belong to the same row, column, or box is not a standard subset in that house.