Techniques

Sudoku Hidden Triples Technique and Examples: Complete Guide

Tuesday, March 18, 2025

Introduction

If you're familiar with hidden pairs, you can build on that Sudoku technique and solve for hidden triples. Hidden triples occur when three cells within the same unit (row, column, or 3×3 block) share the same three repeated candidate numbers, or a subset of those numbers, that no other cells in that unit contain.

Hidden triples are similar to naked triples, except the cells contain other candidate numbers, which may make them hard to spot—hidden—at first. However, once you find this hidden set of triples, you can eliminate all other candidate numbers within these three cells because the cells must contain one of the three candidate options in that triple set.

With this Sudoku strategy, beginners can solve sudoku puzzles faster and advanced solvers can rely on this method to consistently work through more challenging puzzles, eliminating candidates and solving puzzles more efficiently.

What Are Hidden Triples in Sudoku?

Hidden triples are a Sudoku solving technique where three cells within the same unit (row, column, or 3×3 block) share the same three candidate numbers or a subset of those numbers, and these three candidate numbers do not appear in any other cells within that unit.

Key Characteristics:

  • Three cells: Must be within the same unit (row, column, or block)
  • Three candidates: Share the same three candidate numbers or a subset
  • Unique to unit: These candidates don't appear in any other cells in that unit
  • Hidden: Cells contain other candidates, making the triple hard to spot

Why They're Called "Hidden": Unlike naked triples where cells have only three candidates (making them obvious), hidden triples are "hidden" because the cells contain additional candidates that obscure the triple pattern. The triple is hidden among other candidates, requiring careful scanning to identify.

Key Points

Essential facts about hidden triples:

  • Three cells, three candidates: Three cells share three candidate numbers or a subset
  • Unique to unit: The three candidates don't appear in any other cells in that unit
  • Other candidates present: Cells contain additional candidates, making them hidden
  • Elimination power: Once found, eliminate all other candidates from the three cells
  • Chain reactions: Can create cascade eliminations leading to answers
  • Advanced technique: Builds on hidden pairs, essential for intermediate and hard puzzles

How to Find Hidden Triples

Hidden triples differ slightly from the criteria for hidden doubles in that all three candidate numbers may not always appear in all three cells. One cell may simply have a hidden subset of those numbers.

Example: Consider if three cells in a 3×3 block have the following candidates:

  • Cell 1: 2, 3, 5, 7
  • Cell 2: 3, 5, 7
  • Cell 3: 1, 5, 7

As long as no other cells in that 3×3 block have a 3, 5, or 7, you've found a hidden triple because the third cell contains a subset (5, 7) of the triple (3, 5, 7). Since these candidates don't appear in any other cell, only those three cells can possibly hold a 3, 5, or 7.

Criteria for Hidden Triples

Hidden triples must fit the following criteria:

1. Three cells within the same unit share three candidate numbers or a subset:

  • All three cells are in the same row, column, or 3×3 block
  • They share the same three candidate numbers (e.g., 3, 5, 7)
  • Or one or more cells contain a subset of those numbers (e.g., 5, 7)

2. The three candidate numbers do not appear in any other cells within that unit:

  • Verify that 3, 5, and 7 (in the example) don't appear as candidates in any other cells in that unit
  • This uniqueness is essential for the technique to work

3. Other candidates that are not part of the hidden triple appear in at least one of the three hidden triplet cells:

  • Like the example above, the second cell isn't hiding the candidates, but because other candidates are in the first and third cells, this still qualifies as a hidden triple
  • The "hidden" aspect comes from these additional candidates

Step-by-Step Identification Process

To identify hidden triples, follow these steps:

Step 1: Note all possible candidate numbers within each cell.

  • Use candidate mode or pencil marks to mark all possible candidates
  • Ensure all candidates are marked before scanning

Step 2: Scan rows, columns, and blocks for three cells with the same three candidate numbers or a subset of those numbers.

  • Look for three cells that share three candidate numbers
  • Remember that subsets are allowed (e.g., one cell might have 5, 7 while others have 3, 5, 7)

Step 3: Ensure these numbers don't appear in any other cell within that unit.

  • For example, you find one cell within a row that has 2, 3, 5, and 7 as candidates
  • Another cell has 3, 5, and 7 as candidates
  • A third cell has 1, 5, and 7 as candidates
  • You need to be sure that 3, 5, and 7 are not candidates for any other cells in that row
  • If that's the case, 3, 5, and 7 are the hidden triple

Important Note: Naked quads can contain hidden triples, so remember to scan the whole unit, whether it's a row, column, or block, and look closely at cells that contain multiple candidates.

Hidden Triples Examples

Hidden triples offer a powerful strategy because they may eliminate candidates in the three cells of the hidden triple, help you discover swordfish patterns, or create chain reactions. By eliminating candidates, you narrow down possibilities for other cells, allowing you to find answers as well as eliminate candidates elsewhere in the puzzle.

Here are detailed examples showing how they work in action.

Example 1: Hidden Triples in a Row

In this example, you can find the hidden triple (3, 5, 7) in cells A2, B2, and H2.

The Setup:

  • Cell A2 has candidates: 2, 3, 5, 7
  • Cell B2 has candidates: 3, 5, 7
  • Cell H2 has candidates: 1, 5, 7 (subset of 3, 5, 7)

Verification: Although H2 is a subset (5, 7) of the triple, it contains at least two of the three hidden triple numbers, which makes it part of the hidden triple. Verify it is a hidden triple by seeing if 3, 5, or 7 appear in any other cell in row 2. Because they don't, you've confirmed the hidden triple.

Elimination: Since these three candidate numbers (3, 5, 7) only appear in these three cells, they must be the only answers to those cells. You can remove:

  • The 2 candidate in A2
  • The 1 candidate in H2

Result: After elimination, cells A2, B2, and H2 must contain 3, 5, and 7 in some order, narrowing down the possibilities significantly.

Example 2: Hidden Triples in a Block

This example shows the hidden triple (2, 6, 9) in D1, E2, and F1 of the center block.

The Setup:

  • Cell D1 has candidates: 2, 6, 9, 5
  • Cell E2 has candidates: 2, 6, 9, 1
  • Cell F1 has candidates: 2, 6, 9, 7

Verification: Verify this by checking that 2, 6, or 9 don't appear in any other cells in the middle 3×3 block. Once confirmed, you can eliminate the following candidates:

  • 5 in D1
  • 1 in E2
  • 7 in F1

Chain Reaction: You can see the chain reaction this technique created by looking at cells F2 and D3. Because 1 was eliminated from E2 and 7 was eliminated from F1, you're left with a naked pair (1, 7) in F2 and D3. This allows you to eliminate 7 as a candidate from E3, making 5 the answer for that cell.

Alternative Path: You can arrive at 5 as the answer for E3 in another way. After 5 was eliminated as a candidate from D1, that means E3 is the only possible cell in which 5 could be an answer for this block.

Key Insight: Either way, using hidden triples not only helps eliminate candidates but can also create a cascade of elimination that results in answers for cells. This demonstrates the power of hidden triples in creating chain reactions.

Example 3: Hidden Triples in a Column

In this example, the hidden triple (4, 6, 9) appears in column C.

The Setup:

  • Cell C6 has candidates: 4, 6, 9
  • Cell C8 has candidates: 4, 6, 9, 8
  • Cell C9 has candidates: 4, 6, 9, 1

Verification: Although C8 includes the candidates 4, 6, and 8, it does include a subset of the hidden triple (4, 6), which still makes it part of the hidden triple. Verify that 4, 6, and 9 don't appear in any other cells in column C.

Elimination: Because 4, 6, and 9 must be the answers in cells C6, C8, and C9, you can eliminate:

  • The 8 candidate in C8
  • The 1 candidate in C9

Result: After elimination, cells C6, C8, and C9 must contain 4, 6, and 9 in some order, significantly narrowing down the possibilities in column C.

Hidden Triples vs. Naked Triples

Understanding the difference between hidden and naked triples helps you identify both patterns more effectively.

Naked Triples

Characteristics:

  • Three cells contain only the same three candidates (no other candidates)
  • Easy to spot because cells have only three candidates
  • Obvious pattern visible in candidate marks

Example:

  • Cell 1: 3, 5, 7
  • Cell 2: 3, 5, 7
  • Cell 3: 3, 5, 7

Hidden Triples

Characteristics:

  • Three cells share three candidates but also contain other candidates
  • Hard to spot because cells have multiple candidates
  • Hidden pattern requires careful scanning

Example:

  • Cell 1: 2, 3, 5, 7
  • Cell 2: 3, 5, 7
  • Cell 3: 1, 5, 7

Key Difference: Naked triples are obvious (cells have only three candidates), while hidden triples are hidden (cells have additional candidates). Both eliminate the same candidates from other cells in the unit, but hidden triples also eliminate other candidates from the three cells themselves.

Strategies for Finding Hidden Triples

Use Candidate Mode

Candidate mode (or pencil marks) is essential for finding hidden triples:

  • Mark all candidates: Ensure all possible candidates are marked before scanning
  • Visual scanning: Makes it easier to spot patterns across cells
  • Systematic approach: Scan each unit methodically

Scan Systematically

Develop a systematic scanning routine:

  • Start with rows: Scan each row for hidden triples
  • Then columns: Move to columns after rows
  • Finally blocks: Check 3×3 blocks last
  • Be thorough: Don't skip units or rush through scanning

Look for Subsets

Remember that hidden triples can include subsets:

  • Not all cells need all three: One cell might have only two of the three candidates
  • Subsets are valid: 5, 7 is a valid subset of 3, 5, 7
  • Check carefully: Verify that the three candidates don't appear elsewhere

Verify Uniqueness

Always verify that the three candidates don't appear in other cells:

  • Check the entire unit: Scan all cells in the row, column, or block
  • Confirm uniqueness: The three candidates must be unique to those three cells
  • Double-check: Verification is essential for the technique to work

Common Mistakes to Avoid

Not Verifying Uniqueness

Mistake: Assuming three cells form a hidden triple without checking if the candidates appear elsewhere.

Solution: Always verify that the three candidate numbers don't appear in any other cells in that unit.

Missing Subsets

Mistake: Looking only for cells that contain all three candidates, missing subsets.

Solution: Remember that subsets are valid—one cell might have only 5, 7 while others have 3, 5, 7.

Overlooking Chain Reactions

Mistake: Eliminating candidates but not looking for resulting patterns or answers.

Solution: After eliminating candidates, scan for new patterns like naked pairs or singles that may have been created.

Not Using Candidate Mode

Mistake: Trying to find hidden triples without marking all candidates first.

Solution: Always use candidate mode or pencil marks to mark all possible candidates before scanning for hidden triples.

Advanced Applications

Creating Chain Reactions

Hidden triples often create chain reactions:

  • Eliminate candidates: Remove other candidates from the three cells
  • Look for new patterns: Check if eliminations create naked pairs, singles, or other patterns
  • Follow the chain: Continue solving using newly revealed patterns

Discovering Swordfish Patterns

Hidden triples can help discover swordfish patterns:

  • Look across units: Hidden triples in multiple rows or columns may indicate swordfish
  • Pattern recognition: Understanding hidden triples improves swordfish identification
  • Advanced solving: Combines multiple techniques for complex puzzles

Working with Naked Quads

Naked quads can contain hidden triples:

  • Scan thoroughly: Don't stop at naked quads—check for hidden triples within
  • Multiple patterns: A unit may contain both naked and hidden patterns
  • Comprehensive scanning: Look for all possible patterns in each unit

Summary

Hidden triples are a powerful Sudoku technique where three cells within the same unit (row, column, or 3×3 block) share the same three candidate numbers or a subset of those numbers, and these candidates don't appear in any other cells in that unit.

Unlike naked triples where cells have only three candidates (making them obvious), hidden triples are "hidden" because the cells contain additional candidates that obscure the triple pattern. Once found, you can eliminate all other candidate numbers within these three cells because the cells must contain one of the three candidate options.

The criteria for hidden triples are: (1) three cells within the same unit share three candidate numbers or a subset, (2) the three candidate numbers do not appear in any other cells within that unit, and (3) other candidates (not part of the hidden triple) appear in at least one of the three hidden triplet cells.

To find hidden triples: note all possible candidate numbers within each cell, scan rows/columns/blocks for three cells with the same three candidate numbers or a subset, and ensure these numbers don't appear in any other cell within that unit.

Examples demonstrate hidden triples in rows (3, 5, 7 in cells A2, B2, H2), blocks (2, 6, 9 in D1, E2, F1), and columns (4, 6, 9 in C6, C8, C9), showing how they eliminate candidates and create chain reactions that lead to answers.

Hidden triples offer a powerful strategy because they may eliminate candidates in the three cells, help discover swordfish patterns, or create cascade eliminations that narrow down possibilities and find answers. This technique helps beginners solve faster and advanced solvers work through challenging puzzles more efficiently.

Hidden triples may sound like an advanced technique, but they build off of hidden pairs and can be a reliable technique to unlock answers in intermediate or hard Sudoku puzzles. With practice and systematic scanning, you'll be able to spot hidden triples and use them to solve puzzles more efficiently.

Ready to practice hidden triples? Try our Sudoku game, explore daily challenges, or learn more solving techniques to improve your skills!

❓ FAQ

Q1: What are hidden triples in Sudoku?

Hidden triples occur when three cells within the same unit (row, column, or 3×3 block) share the same three repeated candidate numbers or a subset of those numbers that no other cells in that unit contain. Unlike naked triples where cells have only three candidates, hidden triples are "hidden" because the cells contain additional candidates that obscure the triple pattern.

Q2: How do I find hidden triples?

To find hidden triples: (1) note all possible candidate numbers within each cell using candidate mode or pencil marks, (2) scan rows, columns, and blocks for three cells with the same three candidate numbers or a subset of those numbers, and (3) ensure these numbers don't appear in any other cell within that unit. Verification of uniqueness is essential.

Q3: What's the difference between hidden triples and naked triples?

Naked triples: three cells contain only the same three candidates (no other candidates), making them easy to spot. Hidden triples: three cells share three candidates but also contain other candidates, making them hard to spot. Both eliminate the same candidates from other cells, but hidden triples also eliminate other candidates from the three cells themselves.

Q4: Can hidden triples include subsets of the three candidates?

Yes, hidden triples can include subsets. For example, if three cells share candidates 3, 5, 7, one cell might have only 5, 7 (a subset) while others have 3, 5, 7. As long as the three candidates (3, 5, 7) don't appear in any other cells in that unit, it's a valid hidden triple.

Q5: What candidates can I eliminate when I find a hidden triple?

When you find a hidden triple, you can eliminate all other candidate numbers (not part of the hidden triple) from the three cells that form the hidden triple. For example, if cells have hidden triple (3, 5, 7) but also contain candidates 1, 2, 4, 8, 9, you can eliminate 1, 2, 4, 8, 9 from those three cells.

Q6: Do hidden triples create chain reactions?

Yes, hidden triples often create chain reactions. After eliminating candidates from the three cells, you may discover new patterns like naked pairs or singles. These eliminations can cascade through the puzzle, leading to additional answers and further eliminations, making hidden triples a powerful solving technique.

Q7: Are hidden triples an advanced technique?

Hidden triples are considered an intermediate to advanced technique. They build on hidden pairs and are essential for solving intermediate and hard Sudoku puzzles. While they may sound advanced, with practice and systematic scanning, they become a reliable technique for unlocking answers in challenging puzzles.

Q8: Can naked quads contain hidden triples?

Yes, naked quads can contain hidden triples. When scanning units, remember to look closely at cells that contain multiple candidates. A unit may contain both naked patterns (like naked quads) and hidden patterns (like hidden triples), so comprehensive scanning is important to find all possible patterns.

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