Introduction
Hidden pairs is an intermediate Sudoku technique that identifies situations where two numbers are restricted to only two cells within a row, column, or box. Unlike naked pairs where cells obviously contain only two candidates, hidden pairs are "hidden" because the cells also contain other candidates. Learning to spot hidden pairs significantly improves your ability to solve medium and hard puzzles in Sudoku, making it an essential skill for progressing beyond beginner techniques.
The hidden pairs technique uses logical deduction to identify number restrictions that aren't immediately obvious. When you recognize that two numbers can only appear in two specific cells, you can eliminate all other candidates from those cells, simplifying the puzzle and revealing new solving opportunities.
What Is Hidden Pairs Technique
Hidden pairs is a Sudoku solving technique that identifies when two numbers can only appear in two specific cells within a row, column, or box, even though those cells contain other candidates. The technique is called "hidden" because the pair isn't obvious—the cells appear to have multiple possibilities, but two numbers are actually restricted to just those two cells.
When you identify a hidden pair, you can eliminate all other candidates from those two cells. This elimination is based on the logical fact that if two numbers can only go in two cells, those cells must contain those two numbers, making other candidates impossible. This simplification often reveals naked singles or creates conditions for other techniques.
Hidden pairs work in any region: rows, columns, or boxes. The technique requires scanning each region to identify numbers that can only appear in two cells. This pattern recognition skill improves with practice and becomes essential for solving puzzles beyond the easy difficulty level.
Key Points
Point 1: Hidden Pairs Require Two Numbers Restricted to Two Cells
For a hidden pair to exist, two specific numbers must be able to appear in only two cells within a region. These numbers might have other possible positions initially, but careful analysis reveals they're actually restricted to just two cells. This restriction creates the elimination opportunity.
Point 2: Cells Contain Other Candidates Initially
Unlike naked pairs where cells obviously contain only two candidates, hidden pair cells contain additional candidates. The "hidden" nature comes from these extra candidates obscuring the pair. Eliminating other candidates reveals the pair relationship.
Point 3: Elimination Removes Other Candidates
Once you identify a hidden pair, eliminate all candidates except the two numbers from both cells. This elimination is based on logical necessity: if two numbers can only go in two cells, those cells cannot contain other numbers. This simplification often reveals new solving opportunities.
Point 4: Hidden Pairs Work in All Regions
The technique applies to rows, columns, and boxes equally. Scan each region systematically to find numbers restricted to two cells. The same logical principle works regardless of region type, making hidden pairs a versatile solving tool.
Point 5: Hidden Pairs Often Reveal Other Patterns
After applying hidden pair eliminations, new patterns frequently emerge. Removing candidates may create naked singles, reveal other hidden pairs, or create conditions for advanced techniques. Hidden pairs 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 two cells. This requires checking row, column, and box constraints for each number.
Step 2: Identify Two Numbers with Same Two Cells
When you find a number restricted to two cells, check if another number is also restricted to the same two cells. If two different numbers can only appear in the same two cells, you've found a hidden pair. Verify that both numbers are truly restricted to just those cells.
Step 3: Verify the Hidden Pair
Double-check that both numbers cannot appear elsewhere in the region. Confirm that the two cells are the only possible positions for both numbers. This verification ensures the elimination is valid and prevents errors.
Step 4: Eliminate Other Candidates
Once confirmed, eliminate all candidates except the two numbers from both 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 pair eliminations often create naked singles, reveal other patterns, or simplify candidate notation. Continue the solving process with your updated information.
Examples
Example 1: Hidden Pair in a Row
In row 4, numbers 3 and 7 can only appear in cells R4C2 and R4C8. All other cells in row 4 already contain other numbers or cannot contain 3 or 7 due to column and box constraints. This is a hidden pair. Even though R4C2 contains candidates 2, 3, 5, 7, and 9, and R4C8 contains candidates 3, 4, 7, and 8, we can eliminate 2, 5, and 9 from R4C2, and 4 and 8 from R4C8. This leaves only 3 and 7 in both cells.
Example 2: Hidden Pair in a Box
In box 5, numbers 2 and 6 can only appear in cells R5C5 and R6C5. Checking all cells in box 5, no other cells can contain 2 or 6 due to row and column constraints. This forms a hidden pair. Eliminate all other candidates from R5C5 and R6C5, leaving only 2 and 6. This elimination reveals that R4C5 must be 8, creating further progress.
Example 3: Hidden Pair Revealing Naked Single
After identifying a hidden pair of 4 and 9 in column 7, restricted to cells R2C7 and R8C7, we eliminate other candidates from these cells. This elimination reveals that cell R5C7, which previously had candidates 1, 4, 5, and 9, now cannot contain 4 or 9. With 1 and 5 also eliminated by other constraints, R5C7 becomes a naked single and must be 7.
Summary
Hidden pairs is an essential intermediate Sudoku technique that identifies number restrictions not immediately obvious. By recognizing when two numbers can only appear in two cells, you can eliminate other candidates and simplify puzzles significantly. This technique is crucial for solving medium to hard puzzles where basic methods are insufficient.
Mastering hidden pairs requires practice in pattern recognition and systematic scanning. Learn to scan regions methodically, identifying number restrictions and recognizing when two numbers share the same two-cell restriction. With regular practice, identifying hidden pairs becomes faster and more intuitive.
Practice hidden pairs in your next Sudoku session, then explore more techniques in Sudoku Tips and Strategies: Complete 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 pairs from naked pairs?
Naked pairs are obvious—two cells contain exactly the same two candidates with no other possibilities. Hidden pairs are "hidden" because the cells contain additional candidates, but two numbers are restricted to just those two cells. Hidden pairs require eliminating other candidates to reveal the pair relationship.
Q2: Can hidden pairs work with more than two numbers?
Yes, hidden triples work similarly with three numbers restricted to three cells. The same principle applies: if three numbers can only appear in three cells, eliminate other candidates from those cells. Hidden triples are less common but follow the same logical pattern.
Q3: How often do hidden pairs appear in puzzles?
Hidden pairs appear regularly in medium to hard difficulty puzzles. In medium puzzles, you might find one or two hidden pairs. In hard puzzles, hidden pairs appear more frequently, often multiple times with different number combinations. They're essential for progressing beyond basic techniques.
Q4: Do I need candidate notation to find hidden pairs?
Yes, candidate notation (pencil marks) is essential for finding hidden pairs. Without seeing all possible candidates in cells, identifying number restrictions is nearly impossible. Proper notation makes scanning for hidden pairs much more efficient and accurate.
Q5: What if I find a hidden pair but it doesn't help?
If a hidden pair doesn't immediately reveal new opportunities, continue solving with other techniques. The elimination still simplifies the puzzle and may help later. Sometimes hidden pairs create conditions for other techniques that aren't immediately obvious.
Q6: Should I look for hidden pairs before or after other techniques?
Look for hidden pairs after basic techniques like single candidates and elimination, but before advanced techniques like X-Wing. Hidden pairs are intermediate techniques that bridge basic and advanced methods. They're often necessary for solving medium puzzles and helpful in hard puzzles.
Next Steps
Ready to master hidden pairs? Play Sudoku now and practice identifying hidden pair patterns in medium difficulty puzzles. For more techniques, read Sudoku Tips and Strategies: Complete Guide and 11 Advanced Sudoku Strategies and Examples. Explore other puzzles in Number Puzzle and Word Puzzle.