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Clear out junk files and repair common Windows errorsFree Scan →Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Hidden singles, hidden pairs, and hidden triples are found by tracking digits across an entire Sudoku unit—not by looking only at how many notes appear in one cell. A unit is one row, one column, or one 3×3 box. For each missing digit, list the cells where it can legally go. One possible cell gives a hidden single; two digits restricted to the same two cells give a hidden pair; three digits whose combined locations are exactly three cells give a hidden triple.
These patterns often sit inside cells that still show several candidates. The deduction comes from the digit’s available positions in the unit, so accurate, current candidate notes are essential.
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What “hidden” means in Sudoku
Sudoku notation gives you two complementary ways to reason. A cell-first approach asks how many candidates remain in one square. A unit-first approach asks where a particular digit can go throughout a row, column, or box. Hidden techniques use the second approach.
The word “hidden” does not mean the information is mysterious or that a special guessing rule is involved. It means the useful restriction is concealed by extra candidates written in the affected cells. A cell may list four notes, yet one of those digits can be the only legal place for that digit in the whole unit. Likewise, two cells may each list four notes while two particular digits can occur nowhere else in the unit.
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Before searching, update candidates against all three constraints: the cell’s row, its column, and its 3×3 box. If a note is stale, you can create a pattern that is not real or miss one that is.
The three patterns at a glance
| Technique | Digits tracked | Cells reserved | What you verify | Immediate result |
|---|---|---|---|---|
| Hidden single | One | One | The digit has exactly one legal position in the unit | Place the digit |
| Hidden pair | Two | Two | Both digits’ complete position lists are contained in the same two cells | Keep only those two digits in those cells; erase other notes |
| Hidden triple | Three | Three | The union of all three position lists is exactly three cells | Keep only the three digits in those cells; erase other notes |
Singles place a value immediately. Pairs and triples usually make eliminations first; those reductions may expose a later single or another pattern in a crossing unit.
How to find a hidden single
Definition
A hidden single is a missing digit that can legally occupy only one empty cell in a particular row, column, or box. The cell can still contain other candidate notes. Because the uniqueness belongs to the digit within the unit, this is different from a naked single, where the cell itself has only one candidate.
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Digit-by-digit scan
- Choose a unit that is not complete: a row, column, or 3×3 box.
- Pick one missing digit.
- Check every empty cell in that unit against its row, column, and box constraints.
- Count the cells where the digit remains legal.
- If the count is one, place that digit in the sole position and update intersecting units.
A practical routine is to sweep one digit through all rows, then all columns, then all boxes, repeating for the next digit. SudokuW recommends this digit-first scan before deeper techniques. It is a useful heuristic, not a promise that every puzzle will produce a placement on every pass.
Example
Imagine a 3×3 box missing several digits. Existing 7s elsewhere in the rows and columns block 7 from every empty square except one. That square may also be marked 2, 5, and 9, but 7 has only one legal position in the box. Place 7 there, then remove 7 from candidates in the square’s row and column.
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How to find a hidden pair
Definition
In one unit, a hidden pair occurs when two missing digits can appear only in the same two cells. Those cells must contain the two digits in either order. Any other notes in those cells are impossible and can be removed.
Position-list method
- Select one row, column, or box and list its missing digits.
- For each digit, record every cell in the unit where it is still legal.
- Look for two digits whose combined locations are exactly two cells.
- Confirm that neither digit has a possible position outside those two cells.
- In both cells, delete every candidate except the two hidden-pair digits.
- Recheck the unit and the rows, columns, or boxes crossing those cells.
Searching by cells is unreliable: a hidden pair does not require either cell to show only two notes at the start. The pair is visible when you compare the positions of digits across the whole unit.
Example
Suppose a row has 4 and 7 missing. The only cells in which 4 is legal are columns 2 and 8, and the only cells in which 7 is legal are also columns 2 and 8. Even if those cells currently read {1,2,4,7} and {3,4,5,7}, they are reserved for 4 and 7. Reduce both to {4,7}; the notes 1, 2, 3, and 5 are eliminated from those cells.
What the pair does not prove
The pair does not tell you which of the two cells contains 4. It also does not directly remove 4 or 7 from unrelated cells in another unit. Its immediate effect is limited to deleting other candidates from the two reserved cells. Those cleaner cells may then create a naked single, a hidden single, or another interaction.
How to find a hidden triple
Definition
A hidden triple is a set of three digits in one unit whose possible positions, taken together, are exactly three cells. The three digits occupy those cells in some order, so every other candidate in those cells can be erased.
The union test
- Choose a unit and identify three missing digits to examine.
- Write the complete position list for each digit within that unit.
- Combine the three lists into one set of cell locations.
- If that union contains exactly three cells, the digits form a hidden triple.
- Check that every listed position for each digit is inside those three cells; one extra position invalidates the deduction.
- Delete all candidates other than the three selected digits from the three reserved cells.
The individual lists do not need to match. One digit might be possible in cells 1 and 4, another in cells 4 and 6, and the third only in cell 6. The union is still cells 1, 4, and 6, so those three cells are reserved for the three digits.
Example
In a row, suppose the legal positions for 4, 5, and 9 together are columns 1, 4, and 6—and nowhere else. The row therefore reserves columns 1, 4, and 6 for {4,5,9}. A cell at column 1 might have notes {1,4,5,8}; after the deduction it becomes {4,5}. A cell at column 4 might become {4,9}, and column 6 might become {5,9}. You do not yet know the exact order, but every other note in those three cells is impossible.
Why “three cells contain three digits” is not enough
Three chosen cells that happen to contain 4, 5, and 9 do not establish a hidden triple. One of those digits may still be legal in a fourth cell elsewhere in the unit. The required proof is positional: all possible locations for the selected digits must be contained in the same three cells.
A reliable workflow during a solve
Use the following sequence whenever ordinary placements slow down. It keeps the deductions auditable and prevents accidental over-elimination.
- Refresh candidates. For every unsolved cell under consideration, remove digits already present in its row, column, or box. Do not rely on notes that were written before recent placements.
- Scan hidden singles. Work digit by digit through incomplete rows, columns, and boxes. Place any digit with exactly one legal position.
- Choose one unit for subsets. Stay inside a single row, column, or box while testing a pair or triple.
- Build position lists. For each missing digit, record every legal cell. A quick notation such as “4: c1,c4” and “5: c4,c6” makes overlaps clear.
- Test the complete unit. For a pair, the union of two lists must be two cells. For a triple, the union of three lists must be three cells. If even one selected digit has an additional legal cell, stop and discard that proposed subset.
- Make only the permitted eliminations. Keep the selected digits in the reserved cells and remove other candidates from those cells. Do not erase the selected digits from unrelated cells.
- Propagate the change. Recheck the affected row, column, and box for new singles or subsets, then continue scanning crossing units.
Keeping candidate notes trustworthy
Update after every placement
When a digit is placed, remove that digit from unsolved cells in the same row, column, and box. A hidden pattern depends on complete position lists; one obsolete note can falsely add a position, while one missing note can hide a real deduction.
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For paper solving, write small candidates in a fixed order and mark deletions cleanly. For a digital grid, use the built-in pencil-mark mode and verify that automatic notes update after a placement. Either method works if the notes represent the current legal possibilities rather than guesses.
Separate proof from speculation
Do not circle a pair or triple because it “looks likely.” Write the positions for each digit and count the union. This small audit is faster than repairing a chain of deductions based on an invalid subset.
Common mistakes and their fixes
- Confusing hidden and naked singles: A hidden single is one position for a digit in a unit; a naked single is one remaining candidate in a cell. Check the unit for the former and the cell for the latter.
- Accepting an incomplete triple: Three cells carrying three selected digits is not proof. Search the entire unit for every legal position of each digit.
- Demanding identical lists for a triple: The three digits may have overlapping but unequal position lists. Only the union must contain exactly three cells.
- Deleting selected digits outside the subset: A hidden pair or triple reserves digits in its cells; it does not directly eliminate those digits from every other cell in the unit.
- Using stale notes: Recalculate candidates after each placement or elimination before trusting a pattern.
- Mixing units: A pair found in one row cannot be justified with positions from a column or box. All positions must be evaluated within one unit.
- Forcing a placement from a pair or triple: The cells are restricted, but their internal order may remain unknown. Wait for another deduction to determine which cell receives which digit.
How to choose what to scan next
Start where the unit is nearly complete
A row, column, or box with only a few missing digits has shorter position lists, so hidden subsets are easier to verify. After each elimination, inspect the same unit before jumping elsewhere; the reduction may immediately reveal a single.
Follow changed intersections
If a hidden pair is found in a box, inspect the two cells’ rows and columns. If a hidden triple is found in a column, inspect the three intersecting boxes. The value of a subset often appears in the crossing units it simplifies.
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Escalate only after basic scans
Run the hidden-single sweep before spending time on triples. Singles produce placements directly, while pairs and triples generally produce notes to remove. This ordering follows the practical digit-first routine described by SudokuW and keeps the search efficient without claiming that every puzzle yields a simple move.
Paper and app solving: a practical checklist
- Choose one unit and write its missing digits.
- For each digit, mark every legal cell with a dot, small note, or position list.
- Circle a digit with one position: hidden single.
- Circle two digits whose locations share exactly two cells: hidden pair.
- Circle three digits whose combined locations are exactly three cells: hidden triple.
- Before erasing, verify the full unit and all constraints affecting each cell.
- Erase only non-subset candidates from reserved cells.
- After every change, refresh notes in the three affected units.
On a screen, zoom or highlight one digit at a time if the interface allows it. On paper, a small position table beside the grid can prevent missed locations when a unit contains many candidates.
Further reading and scope of the terminology
SudokuW’s guides describe the digit-first hidden-single scan and the position-based hidden-pair and hidden-triple tests. Sudoku Mood’s hidden-triple guide, updated 1 September 2026, emphasizes the union-of-positions check, current notes, and follow-up eliminations. Sudoku.com provides a corroborating explanation of hidden pairs. A Wiley-hosted excerpt titled Sudoku Strategies for Advanced Gamers includes material on hidden pairs and hidden triplets; the excerpt does not establish complete bibliographic details, current availability, or pricing.
These techniques are logical deductions, not statistical methods. The consulted technique guides do not provide a named success rate or a study showing how often a given puzzle contains each pattern. Treat the scan order as a practical solving habit, not a guarantee about puzzle difficulty or solution time.
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Quick decision guide
| If you observe | Ask | Action |
|---|---|---|
| A digit is absent from a unit | Where can this digit legally go? | If there is one location, place it as a hidden single. |
| Two digits repeatedly appear in the same two positions | Do either digit have a legal position elsewhere in the unit? | If not, keep only those two digits in those cells. |
| Three digits occupy a combined set of three positions | Is the union exactly three cells, with no extra position? | If yes, keep only those three digits in those cells. |
| A proposed pattern has an extra legal position | Has the subset test actually passed? | No. Recalculate notes and reject the pattern. |
What to remember while solving
Look across a unit for the locations of a digit, rather than judging a cell by its note count. A single location fixes a digit. Two digits confined to two cells form a hidden pair and remove unrelated notes there. Three digits confined by a three-cell union form a hidden triple and remove unrelated notes from those cells. In every case, the proof is the complete, current position list; after each valid elimination, propagate the change through the intersecting rows, columns, and boxes.
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