Interactive Sudoku Technique

Sudoku Pairs, Triples, and Quads: Naked and Hidden Sets

Some useful groups sit in plain sight. Others are buried beneath extra candidate marks. Learning to recognize both kinds helps you clear away candidates without guessing.

The Idea in Plain English

When Digits Belong to Certain Cells, Other Candidates Must Move Aside

Look within one row, one column, or one box. If two digits must fit into the same two cells, those two cells belong to that pair. The same idea works with three digits in three cells and four digits in four cells.

You do not need to know which digit goes into which cell. Knowing that the group must stay together is enough to remove candidates that compete with it.

Pair: two digits sharing two cells Triple: three digits sharing three cells Quad: four digits sharing four cells

Two Ways to Notice a Set

Naked Sets Show the Cells; Hidden Sets Show the Digits

Usually Easy to See

Naked Pair, Triple, or Quad

The cells show only the digits that belong to the group. A Naked Pair often looks like two cells carrying the same two candidates. Since those digits must occupy those cells, remove them from other cells in the same row, column, or box.

Triples and Quads do not need identical candidate marks in every cell. Taken together, the group simply has no room for any digits outside the set.

The Main Focus of This Lesson

Hidden Pair, Triple, or Quad

The important digits appear nowhere else in the row, column, or box, but their cells also contain distracting candidates. Those extra marks hide the set. Remove the distractions from the set's own cells.

Hidden cells do not need to sit beside one another. A pair can be spread across opposite ends of a row or column.

Start Here

How to Find a Hidden Pair

Choose one row, column, or box and scan its candidate marks.
Look for two digits that each appear in only the same two cells.
Ignore the extra candidates written in those two cells.
Keep the pair and remove those extra candidates.

Interactive Example 1 of 3

Remove What Does Not Belong

Select every candidate removed by one complete valid Set or solving route, then check your answer. Selection order does not matter.

0 candidates selected 0 examples mastered

Inspect the whole house before selecting any candidates.

Set digit Your selection Correct removal Missed removal Incorrect selection

One Set Can Reveal Another

The Larger Pattern May Lead You to a Buried Pair

While looking for sets, you may notice that what appears to be a larger set actually exposes a smaller set buried among the other candidates—often a Hidden Pair.

In Row 4, the candidates 3, 5, and 7 attract attention around r4c7, r4c8, and r4c9. They cannot form a Triple because r4c7 also contains 9.

Add r4c4, containing 3 and 9. The four cells r4c4, r4c7, r4c8, and r4c9 now form a Naked Quad using 3, 5, 7, and 9.
The Quad removes 3 and 9 from r4c1, and removes 3 from r4c2.
Both remaining cells are reduced to 6 and 8. The cleanup has exposed a Naked 6–8 Pair—but 6 and 8 were a Hidden Pair in those two cells all along.

Use the larger shape as a clue, not as proof. Before removing anything, confirm that the smaller digits truly appear in only those cells within the row, column, or box.

SudokuParadise grid with a Naked 3-5-7-9 Quad in Row 4 whose eliminations expose a previously hidden 6-8 Pair in r4c1 and r4c2. — The green 3–5–7–9 Quad cleans the red cells in Row 4, exposing the 6–8 Pair that was buried there from the beginning.

Follow One Productive Scan

Three Pairs and a Glimpse of Something More Advanced

The blue 7–9 Pair: In Row 3, r3c1 and r3c4 contain only 7 and 9. Those digits can be removed from the other cells in Row 3.
The green 5–6 Pair: In the same row, r3c7 and r3c9 contain only 5 and 6.
The buried golden 3–8 Pair: After the other two Pairs account for 5, 6, 7, and 9, the remaining unsolved cells r3c3 and r3c6 must contain 3 and 8. Remove 7 from r3c3, and remove 7 and 9 from r3c6.

This is why scanning for sets is not always a search for two identical-looking cells. The two obvious Pairs explain most of the row and uncover the Pair that was hidden beneath extra candidates.

A Preview, Not a Detour

The 5–6 Pair Also Reveals a Simple Unique Rectangle

The Row 3 Pairs and the 5–6 Rectangle are separate deductions discovered during the same scan.

The cells r3c7, r3c9, r9c7, and r9c9 form the corners of a rectangle. Three corners are restricted to 5 and 6. If r9c7 were also allowed to remain 5 or 6, the four corners could swap those digits and leave two solutions.

Because a proper Sudoku has one solution, remove 5 and 6 from r9c7. Its other candidates remain. This simplest form is called a Type 1 Unique Rectangle. The advanced variations deserve their own later lesson.

Important: Unique Rectangle reasoning assumes the puzzle has exactly one solution. It is not needed to explain the three Pairs in Row 3; it is a separate deduction noticed during the same scan.

One scan, five candidate removals: remove 7 from r3c3; remove 7 and 9 from r3c6; and remove 5 and 6 from r9c7.

Keep the Difference Straight

Where Do the Eliminations Go?

What you found	What stays	What you remove
Naked set	The set's digits inside the set's cells	Those digits from other cells in the same house
Hidden set	The hidden digits inside the set's cells	Other digits from the set's own cells

Interactive Check

Set, Useful Clue, or Near Miss?

Choose the best answer. A promising shape can tell you where to look, but only a complete restriction allows eliminations.

1. Buried Across a Row

In Row 6, digits 2 and 8 appear only in r6c2 and r6c8. Both cells also contain other candidates.

Choose an answer.

2. The Third Location

In Box 4, digits 3 and 9 both appear in r4c1 and r5c2, but digit 9 also appears at r6c3.

Choose an answer.

3. Nothing Hidden Here

In Column 5, r2c5 and r9c5 each show only candidates 4 and 7.

Choose an answer.

4. Three Cells, Four Digits

Three cells in a box contain only these combined candidates: 1, 3, 6, and 8. No smaller confirmed set has been found.

Choose an answer.

What You Learned Today

The Technical Name Is a Subset

You have been finding groups of digits restricted to a matching group of cells. More formally, these groups are called subsets. A Pair is a two-cell subset, a Triple uses three cells, and a Quad uses four.

You do not need that terminology to find them. Later, when you study advanced strategies such as Almost Locked Sets, the word subset will describe something you already know how to see.

Common Questions

Pairs, Triples, and Quads Clarified

Must the cells of a set sit next to one another?

No. They only need to share the same row, column, or box. A Hidden Pair can be spread across opposite ends of a line.

Must every digit appear in every cell of a Triple or Quad?

No. The candidates can be distributed unevenly. What matters is that the complete group is restricted to the correct number of cells.

Why are Hidden Pairs harder to find?

Their two cells usually contain additional candidates, so the pair does not look clean. Search for digits that occur in only the same two places instead of searching for identical cells.

Can a Triple or Quad reveal a Pair?

Yes. A larger visible group can help account for several cells and draw attention to two digits restricted to the cells that remain. Confirm the Pair on its own before eliminating anything.

Continue Practicing

Look for Sets in a Complete Puzzle

Generate a fresh Sudoku, enter a puzzle from another source, or review the candidate discipline that makes hidden sets easier to spot.

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