Subsets & Locked Candidates
Naked Quad
A Naked Quad is four cells in one row, column or box whose candidates, pooled together, come to exactly four digits. The four cells must take those four digits between them, so the digits can be removed from every other cell of that unit. No cell has to show all four — two or three each is the usual case — and the idea is the same as a Naked Pair or a Naked Triple, one size bigger.
What you need to see on the grid
Look for a Naked Quad once singles, locked candidates, pairs and triples have stopped producing moves, and only in a unit with eight or nine empty cells: in a fuller unit, the cells left over form a Hidden Single, Pair or Triple that does the same job and is easier to see. Four terms carry the method:
- The candidate union
- Write the candidates of four cells side by side and count the different digits. Exactly four is a Naked Quad; five or more means this group locks nothing. Fewer than four cannot happen on a correct grid — four cells need four different digits — so it means a pencil mark is wrong.
- Partial cells: the 2-2-3-3 shape
- Each cell needs only two to four of the four digits. The example below is {1,2}, {1,2,3}, {1,5} and {1,3,5}, and {1,2}, {2,3}, {3,4}, {1,4} is a quad too. A cell with five candidates can never join; a cell with one is a Naked Single — place it.
- One unit, never two
- A pair or a triple can sit in a row and a box at once and clear both. Four cells cannot: a box holds only three cells of any row or column. A Naked Quad therefore clears exactly one unit.
- The Hidden Quad on the other side
- In a unit with eight empty cells, the four cells outside a Naked Quad are the only places left for the other four digits: that is a Hidden Quad, and it makes the very same eliminations. With n empty cells, the leftover cells always form a hidden subset of n − 4.
How to apply it, step by step
One example runs through all five steps: box 7, in an expert puzzle from the app, at the exact point where its hint needs a Naked Quad. 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. A digit missing from a drawn cell is already placed in its row, column or box, drawn or not. Elsewhere on this board a Locked Candidates move and a Naked Pair are also available; they touch none of the cells drawn here.
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Step 1Pick a unit and shortlist cells with two to four candidates
Choose a row, column or box with eight or nine empty cells and count the candidates in each one. Only cells with two, three or four candidates can belong to a quad: a cell with five already carries more digits than four cells may share.
Box 7 has eight empty cells. R8C2 and R9C2 hold five candidates each and drop out. Six cells stay on the list: R7C2 {1,2}, R7C3 {1,2,3}, R8C1 {1,4,7}, R8C3 {1,5}, R9C1 {1,3,4,6} and R9C3 {1,3,5}.
Only box 7 matters here. Six of its eight empty cells have two to four candidates. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 9 R8 R9 - Digit already on the grid
- Box 7, the unit being searched
- Two to four candidates: can join a quad
- Not needed for this step
Filled in: box 7. 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 cells that pool to exactly four digits
Take four cells from the shortlist, merge their candidates and count the different digits. Any group with R8C1 or R9C1 pools to at least six digits, because their 4, 6 and 7 come on top of the others. Leave both out and the other four cells — R7C2, R7C3, R8C3 and R9C3 — use only 1, 2, 3 and 5: a Naked Quad.
Look at its shape: two, three, two and three candidates. No cell holds all four digits, and that is normal — only the union has to stop at four. Players who wait for a cell reading {1,2,3,5} walk straight past quads like this one.
Technique: Naked Quad
R7C2 {1,2}, R7C3 {1,2,3}, R8C3 {1,5} and R9C3 {1,3,5} use only 1, 2, 3 and 5, and no single cell holds all four. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 9 R8 R9 - Digit already on the grid
- Box 7, the unit being searched
- A cell of the quad
- Candidates 1, 2, 3 and 5, the digits studied
- Not needed for this step
Filled in: box 7. 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 which unit the four cells share
A Naked Quad only clears a unit that contains all four of its cells. Here that is box 7 and nothing else: the cells spread over rows 7, 8 and 9, and three of them share column 3 while R7C2 sits in column 2. It always works out this way, because a box holds only three cells of any row or column — four cells of a box always span at least two rows and two columns, and four cells of a line always spread over at least two boxes.
So rows 7 to 9 and column 3 keep their candidates outside the box: R7C8 still lists 1 and 3, R8C7 lists 1 and 5, R9C8 lists 1, 3 and 5, and the quad says nothing about any of them. If you are used to sweeping a line after a pair or a triple, this is the place to stop.
Technique: Naked Quad
Box 7 with rows 7, 8 and 9. No row holds more than two of the four cells, so the quad lives in box 7 alone. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 9 8 5 6 4 7 R8 3 2 6 9 R9 7 9 2 - Digit already on the grid
- Box 7, the only unit holding all four cells
- A cell of the quad
- Candidates 1, 2, 3 and 5, the digits studied
- Not needed for this step
Filled in: rows 7, 8 and 9; box 7. The other 54 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 the four digits from the rest of the unit
Four cells of one box must hold four different digits, and these four can only hold 1, 2, 3 or 5. So they use up all four digits between them, even though we do not yet know which cell takes which, and no other cell of box 7 can be 1, 2, 3 or 5.
Clear them: R8C1 loses 1; R8C2 loses 1 and 5; R9C1 loses 1 and 3; R9C2 loses 1 and 5. The quad cells keep every candidate.
Now read the same box from the other side. The four other cells are left with {4,7}, {4,7,8}, {4,6} and {4,6,8}: digits 4, 6, 7 and 8 had nowhere else to go in box 7 all along. That is a Hidden Quad, and it removes exactly the candidates you just removed — two views of one fact.
Technique: Naked Quad
Box 7 in full. 1, 2, 3 and 5 leave the four cells outside the quad: seven candidates in all. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 9 R8 R9 - Digit already on the grid
- Box 7, the unit being searched
- A cell of the quad
- Candidates 1, 2, 3 and 5, the digits studied
- Candidate removed
- Not needed for this step
Filled in: box 7. The other 72 cells are left blank: they are not needed for this step. Removed: 1 from R8C1, R8C2, R9C1 and R9C2; 3 from R9C1; 5 from R8C2 and R9C2.
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Step 5Collect what the elimination unlocks
A Naked Quad never places a digit by itself, so look at the units that cross the cells it just cleared. R9C1 sits in column 1, and it lost its 3. Column 1 had two places for 3 — R5C1 and R9C1 — and now only R5C1 is left: a Hidden Single, R5C1 = 3, the app's next hint in this puzzle.
Check every row and column that lost a candidate before you move on. The quad itself lives in one box, but what it unlocks is often found along a line through it.
Technique: Hidden Single
Column 1 after the Naked Quad. R9C1 has lost its 3, so 3 now fits only R5C1. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 2 R3 8 R4 5 R5 3 R6 R7 9 R8 R9 - Digit already on the grid
- Column 1: 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: column 1. The other 72 cells are left blank: they are not needed for this step. Placed: R5C1 = 3.
Common mistakes
Waiting for a cell that shows all four digits
This is how most quads are missed. In the example no cell holds more than three of 1, 2, 3 and 5, and {1,2}, {2,3}, {3,4}, {1,4} has no cell above two. Judge the four cells by their union, never by the longest one.
Sweeping a line the quad does not share
Pairs and triples can lock a line and a box at once, so the habit of clearing both is strong. A quad in a box never shares a line: in the example, R7C3, R8C3 and R9C3 sit in column 3 but R7C2 does not, so removing 2 from R5C3 would be a guess — and a wrong one, because the solution puts 2 in R5C3.
Accepting a union of five digits
One extra candidate in one cell breaks the quad. If R8C3 read {1,5,7}, the four cells would pool to 1, 2, 3, 5 and 7: they could leave one of those digits to the rest of the box, so nothing could be removed. Recount the union whenever a cell changes.
Learn the move, not just the digit
In a real game, when a Naked Quad is what opens the next cell, the hint in the app points to that cell and names the quad in its reasoning — the same move as step 5 above.