Introduction
Hidden triples is an advanced Sudoku technique that extends hidden pairs logic to three numbers and three cells. When three numbers can only appear in three specific cells within a row, column, or box, you can eliminate all other candidates from those cells. Learning to spot hidden triples significantly improves your ability to solve hard puzzles in Sudoku, making it an essential skill for progressing to expert-level solving.
The hidden triples technique uses logical deduction to identify number restrictions that aren't immediately obvious. When you recognize that three numbers are restricted to three specific cells, you can eliminate all other candidates from those cells, simplifying the puzzle and revealing new solving opportunities.
What Is Hidden Triples Technique
Hidden triples is a Sudoku solving technique that identifies when three numbers can only appear in three specific cells within a row, column, or box, even though those cells contain other candidates. The technique is called "hidden" because the triple isn't obvious—the cells appear to have multiple possibilities, but three numbers are actually restricted to just those three cells.
When you identify a hidden triple, you can eliminate all other candidates from those three cells. This elimination is based on the logical fact that if three numbers can only go in three cells, those cells must contain those three numbers, making other candidates impossible. This simplification often reveals naked singles or creates conditions for other techniques.
Hidden triples work in any region: rows, columns, or boxes. The technique requires scanning each region to identify numbers that can only appear in three cells. This pattern recognition skill improves with practice and becomes essential for solving hard and expert puzzles.
Key Points
Point 1: Hidden Triples Require Three Numbers Restricted to Three Cells
For a hidden triple to exist, three specific numbers must be able to appear in only three cells within a region. These numbers might have other possible positions initially, but careful analysis reveals they're actually restricted to just three cells. This restriction creates the elimination opportunity.
Point 2: Cells Contain Other Candidates Initially
Unlike naked triples where cells obviously contain only three candidates, hidden triple cells contain additional candidates. The "hidden" nature comes from these extra candidates obscuring the triple. Eliminating other candidates reveals the triple relationship.
Point 3: Elimination Removes Other Candidates
Once you identify a hidden triple, eliminate all candidates except the three numbers from all three cells. This elimination is based on logical necessity: if three numbers can only go in three cells, those cells cannot contain other numbers. This simplification often reveals new solving opportunities.
Point 4: Hidden Triples Work in All Regions
The technique applies to rows, columns, and boxes equally. Scan each region systematically to find numbers restricted to three cells. The same logical principle works regardless of region type, making hidden triples a versatile solving tool.
Point 5: Hidden Triples Often Reveal Other Patterns
After applying hidden triple eliminations, new patterns frequently emerge. Removing candidates may create naked singles, reveal other hidden patterns, or create conditions for advanced techniques. Hidden triples often act as stepping stones to further progress.
How It Works (Step-by-Step)
Step 1: Scan a Region for Number Restrictions
Choose a row, column, or box to examine. For each number 1-9, identify which cells in that region could contain it. Look for numbers that can only appear in three cells. This requires checking row, column, and box constraints for each number.
Step 2: Identify Three Numbers with Same Three Cells
When you find numbers restricted to three cells, check if three different numbers are all restricted to the same three cells. If three numbers can only appear in the same three cells, you've found a hidden triple. Verify that all three numbers are truly restricted to just those cells.
Step 3: Verify the Hidden Triple
Double-check that all three numbers cannot appear elsewhere in the region. Confirm that the three cells are the only possible positions for all three numbers. This verification ensures the elimination is valid and prevents errors.
Step 4: Eliminate Other Candidates
Once confirmed, eliminate all candidates except the three numbers from all three cells. Remove any other numbers that were possible in those cells. This elimination simplifies the puzzle and often reveals new solving opportunities.
Step 5: Continue Solving
After making eliminations, continue solving using other techniques. Hidden triple eliminations often create naked singles, reveal other patterns, or simplify candidate notation. Continue the solving process with your updated information.
Examples
Example 1: Hidden Triple in a Box
In box 5, numbers 2, 5, and 8 can only appear in cells R4C5, R5C5, and R6C5. All other cells in box 5 already contain other numbers or cannot contain 2, 5, or 8 due to row and column constraints. This is a hidden triple. Even though these cells contain other candidates like 1, 3, 4, 6, 7, and 9, we can eliminate all except 2, 5, and 8 from these three cells.
Example 2: Hidden Triple in a Row
In row 7, numbers 3, 6, and 9 can only appear in cells R7C2, R7C5, and R7C8. Checking all cells in row 7, no other cells can contain 3, 6, or 9 due to column and box constraints. This forms a hidden triple. Eliminate all other candidates from R7C2, R7C5, and R7C8, leaving only 3, 6, and 9. This elimination reveals that R7C4 must be 1, creating further progress.
Example 3: Hidden Triple Revealing Naked Single
After identifying a hidden triple of 1, 4, and 7 in column 3, restricted to cells R2C3, R5C3, and R8C3, we eliminate other candidates from these cells. This elimination reveals that cell R6C3, which previously had candidates 1, 2, 4, 5, 7, and 9, now cannot contain 1, 4, or 7. With 2, 5, and 9 also eliminated by other constraints, R6C3 becomes a naked single and must be 6.
Summary
Hidden triples is an essential advanced Sudoku technique that identifies number restrictions not immediately obvious. By recognizing when three numbers can only appear in three cells, you can eliminate other candidates and simplify puzzles significantly. This technique is crucial for solving hard and expert puzzles where intermediate methods are insufficient.
Mastering hidden triples requires practice in pattern recognition and systematic scanning. Learn to scan regions methodically, identifying number restrictions and recognizing when three numbers share the same three-cell restriction. With regular practice, identifying hidden triples becomes faster and more intuitive.
Practice hidden triples in your next Sudoku session, then explore more techniques in Sudoku Hidden Pairs Technique: Examples and Guide and 11 Advanced Sudoku Strategies and Examples. For different puzzle challenges, try Number Puzzle or Word Puzzle.
FAQ (Frequently Asked Questions)
Q1: How do I distinguish hidden triples from naked triples?
Naked triples are obvious—three cells contain exactly the same three candidates with no other possibilities. Hidden triples are "hidden" because the cells contain additional candidates, but three numbers are restricted to just those three cells. Hidden triples require eliminating other candidates to reveal the triple relationship.
Q2: Can hidden triples work with more than three numbers?
The technique specifically requires three numbers and three cells. However, the same principle extends to hidden quads (four numbers in four cells), though these are extremely rare. Hidden triples are more common and provide significant solving power for hard puzzles.
Q3: How often do hidden triples appear in puzzles?
Hidden triples appear regularly in hard and expert difficulty puzzles. In hard puzzles, you might find one or two hidden triples. In expert puzzles, hidden triples appear more frequently, often multiple times with different number combinations. They're essential for progressing beyond intermediate techniques.
Q4: Do I need candidate notation to find hidden triples?
Yes, candidate notation (pencil marks) is absolutely essential for finding hidden triples. Without seeing all possible candidates in cells, identifying number restrictions is nearly impossible. Proper notation makes scanning for hidden triples much more efficient and accurate.
Q5: What if I find a hidden triple but it doesn't help?
If a hidden triple doesn't immediately reveal new opportunities, continue solving with other techniques. The elimination still simplifies the puzzle and may help later. Sometimes hidden triples create conditions for other techniques that aren't immediately obvious.
Q6: Should I look for hidden triples before or after other techniques?
Look for hidden triples after basic and intermediate techniques like single candidates, elimination, naked pairs, and hidden pairs. Hidden triples are advanced techniques that bridge intermediate and expert methods. They're often necessary for solving hard puzzles and helpful in expert puzzles.
Next Steps
Ready to master hidden triples? Play Sudoku now and practice identifying hidden triple patterns in hard difficulty puzzles. For more techniques, read Sudoku Hidden Pairs Technique: Examples and Guide and 11 Advanced Sudoku Strategies and Examples. Explore other puzzles in Number Puzzle and Word Puzzle.