Subsets & Locked Candidates
Hidden Quad
A Hidden Quad is four digits that, within one row, column or box, can only go in the same four cells. Those four cells must take the four 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 Quad: the cells outside a hidden quad always form a naked set that holds the remaining digits.
What you need to see on the grid
Look for a Hidden Quad once singles, locked candidates, pairs and triples have stopped producing moves, and only in a unit with eight or nine empty cells — with fewer, the naked set on the other side is smaller and easier to find. 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 Quad has two, three or four places: one place is a Hidden Single, and five can never fit into four cells.
- Four digits, four cells
- Put the places of the four digits together: they must come to exactly four cells. No digit has to reach all four, and no cell has to hold all four digits — the only rule is that none of the four digits has a place outside the group.
- The extra candidates
- The other digits pencilled into the four cells. They are what hides the quad — a cell can list seven 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 eight empty cells that set is a Naked Quad; with nine it is five cells and five digits, a set nobody hunts for, and the hidden quad is the way to see it.
How to apply it, step by step
One example runs through all five steps: row 8, part-way through a hard puzzle, with all nine of its cells still empty. 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.
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Step 1Pick a crowded unit and count each digit's places
Choose a row, column or box with eight or nine empty cells. Row 8 here has nine, carrying 41 pencil marks between them. Go through the digits 1 to 9 and note where each one is still a candidate: 1 in C1, C2 and C3; 3 in C1 and C2; 5 in C1 and C5; 6 in C2, C7 and C9; 8 in C1, C2, C3 and C5. Digits 2, 4, 7 and 9 have eight, seven, six and six places.
Keep only the digits with two, three or four places: 1, 3, 5, 6 and 8. A digit with five or more places can never fit into four cells, so 2, 4, 7 and 9 drop out. This count is what makes a hidden quad visible at all — reading the cells one by one, you only see a wall of candidates.
Only row 8 matters here. The kept digits 1, 3, 5, 6 and 8 are highlighted; 2, 4, 7 and 9 have too many places to belong to a quad. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Row being read
- A place for a kept digit
- Candidates 1, 3, 5, 6 and 8, the digits studied
- Not needed for this step
Filled in: row 8. 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 four digits whose places fit in the same four cells
A kept digit with four places names the candidate group at once: 8 sits in C1, C2, C3 and C5. Now check which other kept digits stay inside those cells. 1 (C1, C2, C3), 3 (C1, C2) and 5 (C1, C5) do; 6 does not, because it also reaches C7 and C9. So 1, 3, 5 and 8 have nowhere to go in row 8 except R8C1, R8C2, R8C3 and R8C5. When no digit has four places, merge the places of two or three digits until they cover four cells, and test the rest against that group.
Only 8 reaches all four cells; 3 and 5 reach just two. That is normal: the test is that no digit leaves the group, not that each one fills it. The quad is hidden because the same four cells also carry 2, 4, 6, 7 and 9 — twelve extra candidates in all.
Technique: Hidden Quad
Row 8 in full. 1, 3, 5 and 8 appear only in R8C1, R8C2, R8C3 and R8C5; 6 also reaches R8C7 and R8C9, so it stays out. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Highlighted unit
- A cell of the quad
- Candidates 1, 3, 5 and 8, the digits studied
- Not needed for this step
Filled in: row 8. 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 cells outside the quad: R8C4, R8C6, R8C7, R8C8 and R8C9. Between them they hold only 2, 4, 6, 7 and 9 — five cells, five digits, a naked set. It has to be so, since 1, 3, 5 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 Quad only matters in units with eight or nine empty cells. With eight, the other side is four cells holding four digits — a Naked Quad — and you can use whichever you see first. With seven it is a Naked Triple, with six a Naked Pair, and you will have found those earlier. With nine, as here, it is a set of five that nobody looks for, and the hidden quad is the practical way to see the move.
Row 8 again. The five cells outside the quad list nothing but 2, 4, 6, 7 and 9. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Highlighted unit
- A cell outside the quad
- Candidates 2, 4, 6, 7 and 9, the digits studied
- Not needed for this step
Filled in: row 8. 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 four cells
Row 8 must contain 1, 3, 5 and 8, and each of them can only go into R8C1, R8C2, R8C3 or R8C5. Four digits need four different cells, so these four 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: 2, 4 and 7 from R8C1; 2, 6 and 9 from R8C2; 2, 4, 7 and 9 from R8C3; 2 and 4 from R8C5. The four digits of the quad stay where they are, and the other five cells of the row keep every candidate: 1, 3, 5 and 8 were never there, so the quad has nothing to remove outside its own cells.
Technique: Hidden Quad
Row 8 in full. The twelve extra candidates leave the four quad cells; the rest of the row is untouched. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Highlighted unit
- A cell of the quad
- Candidates 1, 3, 5 and 8, the digits studied
- Candidate removed
- Not needed for this step
Filled in: row 8. The other 72 cells are left blank: they are not needed for this step. Removed: 2 from R8C1, R8C2, R8C3 and R8C5; 4 from R8C1, R8C3 and R8C5; 6 from R8C2; 7 from R8C1 and R8C3; 9 from R8C2 and R8C3.
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Step 5Collect what the cleared cells unlock
A Hidden Quad never places a digit by itself, so look at the units around the cells it just cleared. R8C2 lost its 6. In box 7, 6 had only two places, R8C2 and R9C2; now R9C2 is the only cell left for it, so R9C2 is 6: a Hidden Single.
Do the same for every candidate the quad removed — each one can leave a row, column or box with a single place for that digit. That is where a Hidden Quad pays you back.
Technique: Hidden Single
Box 7 after the Hidden Quad. With 6 gone from R8C2, it fits only R9C2. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 6 - Box 7: one place left for 6
- The cell that must be 6
- Candidate 6, the digit studied
- Digit placed
- Not needed for this step
Filled in: box 7. The other 72 cells are left blank: they are not needed for this step. Placed: R9C2 = 6.
Common mistakes
Expecting every digit in all four cells
In the example only 8 reaches all four cells; 3 and 5 reach two each. A digit with two or three places still counts, as long as they all lie inside the group. Demanding a full spread hides most real quads.
Missing a fifth place
Check all nine cells of the unit for each of the four digits, not only the cells that look promising. Had 6 been taken into the group above, its places in C7 and C9 would have broken it. One overlooked candidate outside the four cells and every deletion inside them is a guess.
Clearing the rest of the unit instead
That is the Naked Quad move, read the wrong way round. The four digits of a hidden quad already have no candidates outside its four cells, so there is nothing to erase there; the move is to delete the other digits inside the four cells.
Learn the move, not just the digit
In a real game, when a Hidden Quad is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 5 above.