Subsets & Locked Candidates
Naked Triple
A Naked Triple is three cells in one row, column or box whose candidates, pooled together, come to exactly three digits. The three cells must take those three digits between them, so the digits can be removed from every other cell of that unit — and of the box as well, when all three cells also sit in one box. No cell has to show all three: {1,2}, {2,3}, {1,3} is a complete triple.
What you need to see on the grid
Look for a Naked Triple once singles, locked candidates and pairs have stopped producing moves, in a unit with six or more empty cells: with five or fewer, the cells left over form a Hidden Pair or a Hidden Single that does the same job and is easier to see. It is the Naked Pair idea one size up, and the Naked Quad is the same idea one size bigger again. Four terms carry the method:
- The candidate union
- Write the candidates of three cells side by side and count the different digits. Exactly three is a Naked Triple; four or more means this group locks nothing. Fewer than three cannot happen on a correct grid — three cells need three different digits — so it means a pencil mark is wrong.
- Partial cells: 2-2-2, 2-2-3, 2-3-3
- Each cell needs only two or three of the three digits. {1,2}, {2,3}, {1,3} is a triple with no cell above two; {1,2,3}, {1,2}, {2,3} and {1,2,3}, {1,2,3}, {1,3} are triples too. A cell with four candidates can never join; a cell with one is a Naked Single — place it.
- A line and a box at once
- A row or column meets a box in exactly three cells. When those three cells are the triple, it lives in both units and clears both. Two cells in a box with the third outside it share the line only, and the box keeps its candidates.
- The Hidden Triple on the other side
- In a unit with six empty cells, the three cells outside a Naked Triple are the only places left for the other three digits: that is a Hidden Triple, and it makes the very same eliminations. With n empty cells, the leftover cells always form a hidden subset of n − 3 — with five, a Hidden Pair, which is easier to spot than the triple it mirrors.
How to apply it, step by step
One board runs through all five steps, part-way through a hard puzzle: row 3 for the first three, row 6 and box 4 for the fourth, box 5 for the payoff. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them.
-
Step 1Pick a unit and shortlist cells with two or three candidates
Choose a row, column or box with six or more empty cells and count the candidates in each one. Only cells with two or three candidates can belong to a triple: a cell with four already carries more digits than three cells may share.
Row 3 has six empty cells. R3C1 {1,2,6,9} and R3C4 {2,3,5,6} hold four candidates each and drop out. Four cells stay on the list: R3C3 {1,9}, R3C5 {5,9}, R3C6 {3,5,6} and R3C9 {1,5}. No two of them show the same pair, so there is no Naked Pair to take first.
Only row 3 matters here. Four of its six empty cells have two or three candidates. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 7 4 8 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 3, the unit being searched
- Two or three candidates: can join a triple
- Not needed for this step
Filled in: row 3. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
-
Step 2Find three cells that pool to exactly three digits
Take three cells from the shortlist, merge their candidates and count the different digits. With four cells there are only four groups to try. Any group with R3C6 pools to five digits, because its 3 and 6 come on top of the others. Leave R3C6 out and the other three cells — R3C3 {1,9}, R3C5 {5,9} and R3C9 {1,5} — use only 1, 5 and 9: a Naked Triple.
Look at its shape: two candidates in each cell, and no cell holds all three digits. This 2-2-2 triple is the one players miss most often. To spot it, follow the pairs round: 1–9, then 9–5, then 5–1, and you are back where you started. Three two-candidate cells that close a loop like this are always a triple.
Technique: Naked Triple
R3C3 {1,9}, R3C5 {5,9} and R3C9 {1,5} use only 1, 5 and 9, and no single cell holds all three. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 7 4 8 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 3, the unit being searched
- A cell of the triple
- Candidates 1, 5 and 9, the digits studied
- Not needed for this step
Filled in: row 3. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
-
Step 3Remove the three digits from the rest of the unit
Three cells of one row must hold three different digits, and these three can only hold 1, 5 or 9. So they use up all three digits between them, even though we do not yet know which cell takes which, and no other cell of row 3 can be 1, 5 or 9.
Clear them: R3C1 loses 1 and 9; R3C4 loses 5; R3C6 loses 5. The triple cells keep every candidate. They sit in three different boxes, so row 3 is the only unit this triple clears.
Now read the same row from the other side. The three other cells are left with {2,6}, {2,3,6} and {3,6}: digits 2, 3 and 6 had nowhere else to go in row 3 all along. That is a Hidden Triple, and it removes exactly the candidates you just removed — two views of one fact.
Technique: Naked Triple
Row 3 in full. 1, 5 and 9 leave the three cells outside the triple: four candidates in all. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 7 4 8 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 3, the unit being searched
- A cell of the triple
- Candidates 1, 5 and 9, the digits studied
- Candidate removed
- Not needed for this step
Filled in: row 3. The other 72 cells are left blank: they are not needed for this step. Removed: 1 from R3C1; 5 from R3C4 and R3C6; 9 from R3C1.
-
Step 4When the three cells also share a box, clear the box too
Three cells can share two units at once. A row or column meets a box in exactly three cells, and when those three cells form the triple, it lives in the line and in the box — so it clears both.
Row 6 on the same board has one. R6C1 {2,3,9}, R6C2 {3,9} and R6C3 {2,9} pool to 2, 3 and 9 — a 2-2-3 shape this time, with one cell holding all three digits and the other two holding two each. All three cells lie in row 6 and in box 4. In row 6, R6C6 loses its 3. In box 4, R5C1 and R5C2 lose their 3 as well: three eliminations where the row alone gives one.
Only 3 goes, because 2 and 9 appear nowhere else in either unit. That is normal: a triple removes whatever its digits still reach, sometimes a lot, sometimes a single candidate.
Technique: Naked Triple
Row 6 with box 4. The triple fills the three cells where they cross, so 3 leaves R6C6 in the row and R5C1, R5C2 in the box. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 4 8 R5 7 R6 8 4 7 R7 R8 R9 - Digit already on the grid
- Row 6 and box 4: each holds all three cells
- A cell of the triple
- Candidates 2, 3 and 9, the digits studied
- Candidate removed
- Not needed for this step
Filled in: row 6; box 4. The other 66 cells are left blank: they are not needed for this step. Removed: 3 from R5C1, R5C2 and R6C6.
-
Step 5Collect what the eliminations unlock
A Naked Triple never places a digit by itself, so look at the units that cross the cells it just cleared. R6C6 sits in box 5 and has lost its 3. Box 5 had two places for 3 — R5C5 and R6C6 — and now only R5C5 is left: a Hidden Single.
Row 5 reaches the same answer through the box bonus. Its other places for 3 were R5C1 and R5C2, and step 4 cleared both. Skip the box sweep and you would see only half of this, so check every row, column and box that lost a candidate before moving on.
Technique: Hidden Single
Box 5 and row 5 after both triples. R6C6, R5C1 and R5C2 have lost their 3, so 3 now fits only R5C5. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 7 9 3 2 4 8 R6 8 4 R7 R8 R9 - Digit already on the grid
- Box 5: one place left for 3
- The cell that must be 3
- Candidate 3, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 5; box 5. The other 66 cells are left blank: they are not needed for this step. Placed: R5C5 = 3.
Common mistakes
Waiting for a cell that shows all three digits
This is how most triples are missed. Row 3 in the example is {1,9}, {5,9}, {1,5}: no cell holds more than two of its three digits. Judge the three cells by their union, never by the longest one.
Clearing a box that does not hold all three cells
The box sweep needs all three cells inside the box. The row-3 triple spreads over boxes 1, 2 and 3, so removing 1 or 9 from R1C1, next to R3C3 in box 1, would be a guess, not a deduction. Two cells in a box are not enough either: only a triple like the one in row 6, with all three cells in box 4, clears its box.
Accepting a union of four digits
One extra candidate in one cell breaks the triple. If R3C9 read {1,5,6}, the three cells would pool to 1, 5, 6 and 9: they could leave one of those digits to the rest of the row, so nothing could be removed. Recount the union whenever a cell changes, or look for a fourth cell and a Naked Quad.
Learn the move, not just the digit
In a real game, when a Naked Triple is what opens the next cell, the hint in the app points to that cell and names the triple in its reasoning — the same move as step 5 above.