Naked and Hidden Quads
Naked and Hidden Quads are the four-cell version of the subset idea you already use for pairs and triples. Four cells in one unit claim four digits between them, and everything else in that unit has to give those digits up. Hidden quads in Sudoku show as four digits confined to four cells; naked quads show as four cells holding four digits.
What the pattern is
A naked quad is four cells in a row, column or box whose candidate lists, taken together, contain exactly four different digits. Those four cells must use up all four digits, so no other cell in the unit can have any of them.
A hidden quad is the same fact seen from the other side: four digits that can appear in only four cells of the unit. Those four cells must swallow all four digits, so every other candidate inside them is dead.
| Naked quad | Hidden quad | |
|---|---|---|
| You count | Digits across four chosen cells | Cells available to four chosen digits |
| Trigger | Four cells, four digits in total | Four digits, four possible cells in total |
| Eliminations | Those digits leave the rest of the unit | Every other digit leaves those four cells |
| Cell sizes | Two, three or four candidates each | Any size, often five or more |
Neither cell needs all four digits. A quad built from {1,4}, {3,8}, {1,3,4} and {4,8} is perfectly valid as long as the union is exactly four digits.
When to look for it
- A unit has enough unsolved cells that something is left to clean up after the quad, and nothing simpler works.
- You can see two or three cells sharing a small digit set, and one more cell would close the group.
- A unit is dense with candidates but four digits keep appearing in the same handful of cells. That is the hidden form.
How to apply it, step by step
- Choose one unit and list the candidates of every unsolved cell.
- For the naked form, take four cells whose combined candidates number exactly four digits.
- Strike those four digits from every other cell in the unit.
- For the hidden form, count the cells available to each digit, then find four digits whose cells add up to exactly four.
- Delete every other candidate inside those four cells and leave the rest of the unit alone.
A worked example
Naked quad
Look at row J: eight cells are empty and only J9 (8) is filled. Watch the four highlighted cells: J1 holds 4 and 9, J4 holds 1, 4 and 5, J5 holds 1, 5 and 9, and J6 holds 4, 5 and 9.
Together the four cells use only 1, 4, 5 and 9. No single cell has all four, but four cells with four digits must hold one of each in some order. So row J cannot use those digits anywhere else. The other cells are J2 (3, 4, 6, 7, 9), J3 (2, 3, 6, 7, 9), J7 (2, 3, 4) and J8 (2, 3, 7). Remove 4 from J2 and J7, and 9 from J2 and J3: four eliminations. J8 holds none of the four digits, so it is untouched.
Hidden quad
A hidden quad is easiest to see by counting cells per digit. Look at box 9, where all nine cells are still empty. Watch where 2, 3, 5 and 7 can go.
In box 9 the digits 2, 3, 5 and 7 fit only in H8, H9, J8 and J9: 2 only in J8 and J9, 3 only in H8 and H9, 5 and 7 in all four. Four digits need four different cells, and only four cells are available, so H8, H9, J8 and J9 hold exactly 2, 3, 5 and 7 in some order. Every other candidate in those cells is deleted: 1 from J8 and J9, 8 from H9 and J9, 9 from H8. The rest of the box stays as it was.
Common mistakes
- Counting a digit twice. The union of the four cells must be four digits, not four appearances.
- Including a cell with a fifth digit "that will probably go away". It either belongs now or it does not.
- Applying a naked quad to cells that are not all in the same unit. Four cells spread over two boxes prove nothing.
- Stripping the wrong side. Naked quads clear the rest of the unit; hidden quads clear the quad cells themselves.
Quintuplets and sextuplets
The naked-set idea does not stop at four cells. Five cells sharing only five digits between them are a naked quintuplet; six cells sharing six digits are a naked sextuplet. But a unit only has nine cells, so a naked set of five or six digits always leaves a smaller complementary hidden set behind (at most four cells), and the hidden version is almost always easier to spot, since fewer cells means a shorter candidate list to scan. That is why quintuplets and sextuplets rarely get named in solving guides: a solver reaches the smaller hidden set first and never needs the large naked one.
Where you meet it
Quads become necessary on hard classic Sudoku and above. They are rarer than pairs and triples, and most quads you find will already have been solved by something cheaper, so reach for one only after singles, pairs, triples and the pointing rules have stopped producing.