Subsets & Locked Candidates
Hidden Triple
A Hidden Triple is three digits that, within one row, column or box, can only go in the same three cells. Those three cells must take the three digits, so every other candidate in them is removed, while the rest of the unit is left as it is. It is the mirror image of a Naked Triple: the cells outside a hidden triple always form a naked set that holds the remaining digits.
What you need to see on the grid
Look for a Hidden Triple once singles, locked candidates and pairs have stopped producing moves, in a unit with seven or more empty cells — with six, the other side is a Naked Triple that does the same job, and with fewer it is a pair or a single you would have found already. Four terms carry the method:
- Places of a digit
- For a digit still missing from the unit, the cells where it is still a candidate. A digit of a Hidden Triple has two or three places: one place is a Hidden Single, and four can never fit into three cells.
- Three digits, three cells
- Put the places of the three digits together: they must come to exactly three cells. No digit has to reach all three, and no cell has to hold all three digits — the only rule is that none of the three digits has a place outside the group.
- The extra candidates
- The other digits pencilled into the three cells. They are what hides the triple — a cell can list five candidates and still be part of one — and they are exactly what the technique erases.
- The naked set on the other side
- The unit's other empty cells hold only the remaining digits, so they form a naked set of their own, and it removes the same candidates. With six empty cells that set is a Naked Triple; with seven, a Naked Quad; with eight or nine it is five or six cells, a set nobody hunts for, and the hidden triple is the way to see it.
How to apply it, step by step
One example runs through all five steps: row 2 as you might meet it part-way through a hard puzzle, with a 9 already placed and eight empty cells. 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. The pencil marks are what the digits in the blank rows, columns and boxes leave — you do not need to see them to follow the step.
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Step 1Read a unit digit by digit and note where each one can go
Choose a row, column or box with seven or more empty cells. Row 2 here has eight, with 28 pencil marks between them. Instead of reading the cells one by one, go through the missing digits 1 to 8 and note where each one is still a candidate: 1 in C2 and C8; 6 in C2, C4 and C8; 7 in C2, C6 and C7; 8 in C4 and C8. Digits 2, 3, 4 and 5 have five, four, four and five places.
Keep only the digits with two or three places: 1, 6, 7 and 8. A digit with four or more places can never fit into three cells, so 2, 3, 4 and 5 drop out. While you count, check for the simpler moves: no digit has a single place, and the two digits with two places, 1 and 8, do not share the same two cells, so there is no Hidden Single or Hidden Pair to take first.
Only row 2 matters here. The kept digits 1, 6, 7 and 8 are highlighted; 2, 3, 4 and 5 have too many places to belong to a triple. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 9 R3 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 2, the unit being read
- A place for a kept digit
- Candidates 1, 6, 7 and 8, the digits studied
- Not needed for this step
Filled in: row 2. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Find three digits whose places fit in the same three cells
A kept digit with three places names a candidate group at once: 6 sits in C2, C4 and C8. Now check which other kept digits stay inside those cells. 1 (C2, C8) and 8 (C4, C8) do; 7 does not, because it also reaches C6 and C7. So 1, 6 and 8 have nowhere to go in row 2 except R2C2, R2C4 and R2C8. If 7's three places had named the group instead, 1 and 8 would have fallen outside it at once — try each three-place digit in turn, or merge the places of two two-place digits, and test the rest against that group.
Only 6 reaches all three cells; 1 and 8 reach two each. That is normal: the test is that no digit leaves the group, not that each one fills it. The triple is hidden because the same three cells also carry 2, 3, 4, 5 and 7 — six extra candidates in all.
Technique: Hidden Triple
Row 2 in full. 1, 6 and 8 appear only in R2C2, R2C4 and R2C8; 7 also reaches R2C6 and R2C7, so it stays out. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 9 R3 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 2, the unit being read
- A cell of the triple
- Candidates 1, 6 and 8, the digits studied
- Not needed for this step
Filled in: row 2. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Check the other side: the remaining cells form a naked set
Look at the five empty cells outside the triple: R2C1, R2C3, R2C6, R2C7 and R2C9. Between them they hold only 2, 3, 4, 5 and 7 — five cells, five digits, a naked set. It has to be so, since 1, 6 and 8 are not candidates there. Every hidden subset comes with a naked one on the other side of its unit, and the two remove exactly the same candidates.
This is why a Hidden Triple only matters in units with seven or more empty cells. With six, the other side is three cells holding three digits — a Naked Triple — and you can use whichever you see first. With five it is a Naked Pair, with four a Naked Single, and you will have found those earlier. With seven it is a Naked Quad, and with eight, as here, a set of five that nobody looks for: the hidden triple is the practical way to see the move.
Row 2 again. The five empty cells outside the triple list nothing but 2, 3, 4, 5 and 7. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 9 R3 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 2, the unit being read
- An empty cell outside the triple
- Candidates 2, 3, 4, 5 and 7, the digits studied
- Not needed for this step
Filled in: row 2. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 4Remove every other candidate from the three cells
Row 2 must contain 1, 6 and 8, and each of them can only go into R2C2, R2C4 or R2C8. Three digits need three different cells, so these three cells are used up by them — even though we do not know yet which digit goes where — and no other digit can sit there.
Erase the extras: 4 and 7 from R2C2; 2 and 3 from R2C4; 2 and 5 from R2C8. The three digits of the triple stay where they are, and the other cells of the row keep every candidate: 1, 6 and 8 were never there, so the triple has nothing to remove outside its own cells. Once cleared, the three cells read {1,6}, {6,8} and {1,6,8} — the hidden triple has become a naked one.
Technique: Hidden Triple
Row 2 in full. The six extra candidates leave the three triple cells; the rest of the row is untouched. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 9 R3 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row 2, the unit being read
- A cell of the triple
- Candidates 1, 6 and 8, the digits studied
- Candidate removed
- Not needed for this step
Filled in: row 2. The other 72 cells are left blank: they are not needed for this step. Removed: 2 from R2C4 and R2C8; 3 from R2C4; 4 from R2C2; 5 from R2C8; 7 from R2C2.
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Step 5Collect what the cleared cells unlock
A Hidden Triple never places a digit by itself, so look at the units around the cells it just cleared. R2C2 lost its 7. In box 1, 7 had only two places, R2C2 and R3C1; now R3C1 is the only cell left for it, so R3C1 is 7: a Hidden Single. Box 1 had no single before the triple — this one is new.
Do the same for every candidate the triple removed — each one can leave a row, column or box with a single place for that digit. The 4 that left R2C2 changes nothing here, for instance, because box 1 still has two places for 4: R1C2 and R2C3. That search is where a Hidden Triple pays you back.
Technique: Hidden Single
Box 1 after the Hidden Triple. With 7 gone from R2C2, it fits only R3C1. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 9 R2 R3 7 8 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Box 1: one place left for 7
- The cell that must be 7
- Candidate 7, the digit studied
- Digit placed
- Not needed for this step
Filled in: box 1. The other 72 cells are left blank: they are not needed for this step. Placed: R3C1 = 7.
Common mistakes
Expecting every digit in all three cells
In the example only 6 reaches all three cells; 1 and 8 reach two each, and no cell holds all three digits except R2C8. A digit with two places still counts, as long as they lie inside the group. Demanding a full spread hides most real triples.
Missing a fourth place
Check every cell of the unit for each of the three digits, not only the cells that look promising. 7 shares R2C2 with the triple and has three places, just like 6 — but two of them, R2C6 and R2C7, lie outside the group. Read only the cells around R2C2 and 7 looks like a member. One overlooked candidate outside the three cells and every deletion inside them is a guess.
Clearing the rest of the unit instead
That is the Naked Triple move, read the wrong way round. The three digits of a hidden triple already have no candidates outside its three cells, so there is nothing to erase there; the move is to delete the other digits inside the three cells.
Keep going
- Hidden PairTwo digits with only the same two cells left in a unit clear every other candidate from those cells.
- Naked TripleThree cells that pool to exactly three digits clear those digits from the rest of their unit.
- Hidden QuadFour digits with only four cells to go to in a unit clear every other candidate from those cells.
Learn the move, not just the digit
In a real game, when a Hidden Triple is what opens the next cell, the hint in the app points to that cell and explains why the digit fits nowhere else, the triple's eliminations included — the same move as step 5 above.