SuuudokuuuNaked Quad Sudoku Technique
Updated
A naked quad is four cells in the same unit whose candidates, taken together, use only four different digits, which reserves those four digits for those four cells and removes them from every other cell of the unit.
When a naked quad applies
Naked quads need room. A unit has to keep at least six empty cells for a quad to be worth anything, because four of them form the quad and there must still be somebody left to eliminate from. That makes quads a mid-solve technique on hard puzzles rather than something you will meet on an easy grid.
The counting argument is the same one that powers every naked subset. Four cells that between them can only hold four digits will use up all four, one per cell. The unit contains each digit once, so those digits are unavailable to the rest of it.
In practice quads are frequently spotted the other way around. If a unit has six blanks and two of them form a hidden pair, the other four automatically form a naked quad. Recognising the complement is often faster than testing combinations by hand.
Worked example
Row 3 has six empty cells. Four of them share a narrow set of options: r3c1 holds 7, 8 and 9; r3c7 holds 6, 7, 8 and 9; r3c8 holds 7, 8 and 9; r3c9 holds 6, 7 and 9. Merging those lists gives exactly 6, 7, 8 and 9, so the quad is genuine.
The remaining blanks are r3c2 and r3c5. r3c2 already held only 1 and 2, so it is untouched. r3c5 held 1, 2, 6, 7, 8 and 9, and loses four candidates in one stroke, leaving 1 and 2. Row 3 now has a naked pair on 1 and 2, and the row has effectively split into two closed groups.
How to spot a naked quad
- Find a unit with six or more blanksQuads need space. A unit with five blanks or fewer cannot produce a useful quad elimination.
- Discard cells with five or more candidatesNo cell in a quad can hold more than four candidates, so any wider cell is out of the running straight away.
- Try the complement firstIf two cells in the unit form a hidden pair, or if the unit has exactly six blanks and two of them share a tight pair, the other four are your quad.
- Merge four candidate lists and countA union of exactly four digits confirms the quad. Then remove all four digits from every other cell of the unit.
Common mistakes
- Accepting a union of five digits. Four cells covering five digits carry no information.
- Testing every four-cell combination by brute force. There are far too many; filter by candidate count first and look at the complement.
- Announcing a quad in a unit with only five blanks. The pattern may be real but there is at most one cell left to eliminate from, and it is usually already solved by a simpler technique.
- Erasing candidates from the quad cells. As with every naked subset, the quad keeps everything it had.
Naked quad FAQ
Do all four cells need four candidates?
No. Cells with two or three candidates are welcome members of a quad. The only rule is that the union across the four cells contains exactly four digits.
Is a naked quad worth looking for?
Only after pairs, triples and the intersection techniques have been exhausted. A quad usually implies a simpler hidden subset elsewhere in the same unit, and that simpler pattern is quicker to find.
How does a naked quad relate to a hidden pair?
They are complements. In a unit with six empty cells, a naked quad on four of them is exactly a hidden pair on the other two, and either name describes the same underlying deduction.
Is there a naked quint?
In principle yes, but a five-cell subset in a nine-cell unit is always the complement of a four-cell hidden subset, so solvers stop at four. Suuudokuuu implements naked and hidden subsets up to size four for the same reason.