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A Hidden Triple is a three-digit Hidden Subset.
Inside one row, column, or box, choose three candidate digits. If every possible position for those three digits is confined to the same three unsolved cells, then those cells are reserved for the three digits.
The cells may contain many distracting candidates.
Those extra candidates can be removed.
The central idea is:
3 digits are restricted to 3 cells.
That is the Hidden Pair rule scaled from two digits to three.
Quick rule
A Hidden Triple requires:
- three target digits in one row, column, or box;
- every legal position for those digits in that house to lie inside the same three cells;
- no target digit to have a legal position elsewhere in the source house.
Then:
keep the three target digits in the three cells and remove every unrelated candidate from those cells.
Schematic unit view — crossed candidates are the logical eliminations.
- Source candidates
- Eliminations
A simple Hidden Triple
Suppose one box contains three important cells:
- A
{1,2,5,7} - B
{2,3,5,8} - C
{4,5,6,8}
Now search the entire box by digit, not by cell.
Suppose:
- candidate 2 appears only in A/B;
- candidate 5 appears only in A/B/C;
- candidate 8 appears only in B/C;
- none of 2, 5, or 8 appears anywhere else in the box.
The three digits {2,5,8} are confined to three cells.
Therefore A/B/C must eventually contain 2, 5, and 8 in some order.
The unrelated candidates are impossible and can be removed.
The cells become:
- A
{2,5} - B
{2,5,8} - C
{5,8}
The previously hidden reservation is now visible.
The target digits do not need to appear in all three cells
A Hidden Triple does not require this:
- A
{2,5,8} - B
{2,5,8} - C
{2,5,8}
That is only one possible pattern.
A target digit may appear in:
- two of the three cells;
- all three cells.
For example:
- 2 → A/B
- 5 → A/B/C
- 8 → B/C
is perfectly valid.
What matters is not equal candidate sets.
What matters is that the three digits have no legal position outside A/B/C in the source unit.
Why Hidden Triples work
The row, column, or box still needs one copy of each target digit.
So it must place three required digits: X, Y, and Z.
If all positions for X/Y/Z are contained in only three cells, those cells are fully occupied by the three target digits.
Suppose one of those cells instead took an unrelated candidate W.
Only two cells would remain available for X, Y, and Z.
Three required digits cannot fit into two cells.
Therefore W cannot be true.
That is the complete proof.
The exact arrangement of X/Y/Z remains unknown, but the exclusion of unrelated candidates is certain.
Step-by-step board example
Hidden Triples can appear in any house
The source unit may be:
- a row;
- a column;
- a box.
The three cells do not need to be adjacent.
The important boundary is the unit in which the three target digits are being counted.
A candidate occurrence elsewhere in another overlapping house does not matter unless it is also inside the source house.
This is why candidate-position scanning must always answer:
Which row, column, or box am I proving this Hidden Triple in?
Hidden Triple vs Hidden Pair
Hidden Subsets scale by the number of digits and cells.
Hidden Pair
2 digits → confined to 2 cells.
Hidden Triple
3 digits → confined to 3 cells.
Hidden Quad
4 digits → confined to 4 cells.
The proof stays the same.
The search gets harder because more candidate-position maps must be compared at once.
Hidden Triple vs Naked Triple
These are dual descriptions of subset logic.
Hidden Triple
Start with digits:
Where can X, Y, and Z go in this unit?
If all of their positions are inside the same three cells, remove other candidates from those cells.
Naked Triple
Start with cells:
Which digits can these three cells contain?
If the union is exactly X/Y/Z, remove X/Y/Z from other cells in the unit.
After a Hidden Triple is cleaned, the three cells contain only the reserved digits, so the resulting candidate state can also be described from the Naked viewpoint.
The final reservation is the same.
The discovery method is not.
Candidate positions are the key
Hidden Triples are much easier to understand when you stop reading candidate lists cell by cell.
Instead, choose a digit and ask:
Where can this digit go in this unit?
Do that for three digits.
For example:
digit 2 → A/B
digit 5 → A/B/C
digit 8 → B/CThe union of positions is {A,B,C}.
Three digits.
Three possible cells.
That is the Hidden Triple test.
For Naked Triples you union candidate digits across cells.
For Hidden Triples you union candidate positions across digits.
That symmetry is one of the most useful ways to remember the difference.
How to find Hidden Triples efficiently
Do not search by comparing every messy three-cell group.
Search digit-first.
1. Choose one unit
Rows, columns, and boxes with several unsolved cells are natural targets.
2. Find restricted candidate digits
Look for digits that appear only two or three times in that house.
A digit with six possible positions is unlikely to be useful in a Hidden Triple search.
3. Compare three restricted digits
Overlay their position sets mentally or with candidate highlighting.
4. Count the union of positions
If all positions for the three digits lie inside exactly three cells, you have the subset.
If the union spans four or more cells, those three digits do not form a Hidden Triple.
5. Remove unrelated candidates
Only candidates other than the three target digits are eliminated, and only from the three reserved cells.
6. Re-scan the cleaned cells
The cleanup may create another immediate deduction.
A useful validity checklist
Before removing candidates, verify:
- Are you working inside one specific row, column, or box?
- Have you checked all legal positions for each of the three target digits in that house?
- Is the union of those positions exactly three cells?
- Are the candidates you plan to remove unrelated to the target three digits?
- Are your candidate notes complete enough for positional counting?
The fifth question is especially important in manually notated puzzles.
A missing legal candidate can create a false Hidden Triple.
Why Hidden Triples feel harder than Naked Triples
The logic is not significantly more advanced.
The recognition burden is.
A Hidden Triple can be difficult to see because:
- the three cells may contain many notes;
- each target digit can occupy a different subset of the cells;
- the pattern is defined by absence elsewhere in the unit;
- three separate candidate-position maps must be compared;
- unrelated candidates visually dominate the cells before cleanup.
This is why digit highlighting or candidate filters can be especially useful for Hidden Subsets.
The tool does not provide the proof.
It makes the relevant positional structure easier to see.
What the cleanup can unlock
A Hidden Triple may:
- reduce one cell to a Naked Single;
- create a bivalue cell;
- expose a Naked Pair or Naked Triple;
- create a Locked Candidate in an overlapping house;
- reduce candidate noise enough to reveal another Hidden subset.
After cleanup, inspect each affected cell in:
- its row;
- its column;
- its box.
The most valuable consequence may appear outside the source unit.
Common mistakes
Looking for three identical candidate lists
Hidden Triples are defined by the positions of three digits, not visual equality between cells.
Requiring every target digit in all three cells
Not necessary. Two or three positions per target digit are common.
Removing the target digits
Those are the reserved candidates.
Remove only the unrelated candidates from the three cells.
Missing one target position elsewhere in the unit
If candidate X is also legal in a fourth cell in the source house, the proposed three-cell confinement fails.
Counting incomplete pencil marks as complete candidates
Hidden-subset searches require trustworthy occurrence information.
Searching Hidden Triples before simpler logic
Hidden Singles and Hidden Pairs are easier positional restrictions. Check them first.
Recognition drill
Choose one unit and write three low-frequency candidate maps:
2 → A/B
5 → A/B/C
8 → B/CThen ask:
How many distinct cells contain every possible position for these three digits?
If the answer is three, inspect those cells for removable candidates.
FAQ
Can a Hidden Triple cell contain extra candidates?
Yes. Those extra candidates are exactly what the technique removes.
Does each target digit need exactly three possible positions?
No. A target digit may have two or three positions inside the three-cell set.
Can one target digit have only one position?
If a digit has only one legal position in the house, you already have a Hidden Single. Solve the simpler deduction first.
Can a Hidden Triple become a Naked Triple?
After cleanup, the three cells contain only the three reserved digits, so the same subset can be described from the Naked viewpoint.
Does a valid Hidden Triple always eliminate candidates?
Not necessarily. If the three cells already contain only the target digits, the reservation is true but there is nothing left to clean.
Are Hidden Triples common?
They are less immediately visible than Pairs and may occur less often in ordinary solving paths, but they are a standard subset technique and become much easier to recognize with accurate candidates.
What to learn next
Once Pairs and Triples are comfortable, move into X-Wing and the Fish family, where the same constrained-set thinking is applied across multiple rows and columns for a single digit.