Naked Quads is the four-cell extension of the Naked Pairs and Naked Triples techniques. A Naked Quad occurs when four cells in the same row, column, or box collectively contain exactly four candidate digits, with each cell holding a subset of those four digits. As with Naked Triples, not every cell needs to contain all four candidates -- the defining criterion is that the union of candidates across all four cells produces exactly four distinct digits.
For instance, four cells containing {1, 3}, {1, 4}, {3, 7}, and {4, 7} form a Naked Quad on digits 1, 3, 4, and 7. These four digits must fill those four cells in some arrangement, so 1, 3, 4, and 7 can be eliminated from every other cell in the shared unit. While the individual cells only hold two candidates each, together they lock four digits into four positions.
Naked Quads are rarer than pairs or triples and can be challenging to spot because you must mentally combine four cells and verify that their candidate union has exactly four elements. However, when found, they tend to produce multiple eliminations that significantly advance the puzzle. Practicing this technique strengthens your ability to visualize subset relationships and prepares you for the intersection-based techniques like Pointing Pairs that analyze how box and line constraints interact.
Try It Yourself
Walk through each step of the naked quads technique on a real puzzle. Follow the instructions and try entering the correct value when prompted.
Look at box 1 (rows 0-2, columns 0-2). Identify the empty cells and list their candidates. Notice that several cells have small candidate lists that could form part of a quad.
Step-by-Step Guide
Update all pencilmarks to ensure accuracy across the grid.
Within a row, column, or box, identify cells with two, three, or four candidates.
Try grouping four cells together and compute the union of their candidate sets.
If the union contains exactly four distinct digits, you have found a Naked Quad.
Eliminate those four digits from all other cells in the same unit.
Verify that no solved cells are contradicted by the eliminations.
Check if the eliminations produce any new singles or enable further techniques.
Four parking spaces in a lot are each reserved for some combination of four specific cars. Since those four cars fill exactly those four spaces, no other car can park in any of them.
The generalized subset principle states that if N cells in a unit collectively contain only N distinct candidates, those N digits must fill those N cells. With four cells and four candidates, every permutation of the four digits across the four cells accounts for all possibilities, so no fifth cell in the unit can hold any of those four digits. This is a direct extension of the pigeonhole argument: four items in four slots leave zero surplus for anything outside the group.
When to use: Use Naked Quads when pairs and triples fail to make progress. Look for four cells in a unit whose combined candidates produce exactly four distinct digits. This is rare but very powerful when found.
Common Mistakes to Avoid
Trying to find Naked Quads before checking for simpler patterns. A Naked Quad might actually contain a Naked Pair or Triple that is easier to spot.
Always scan for Naked Pairs and Triples first. Only look for quads if simpler subsets are not present. Sometimes what appears to be a quad is really a pair plus two resolved cells.
Miscounting the union of candidates across four cells, accidentally including or excluding a digit.
Write out the candidates for each of the four cells and carefully merge them. The union must have exactly four distinct digits -- no more, no less.
Forgetting that cells in a Naked Quad can have two, three, or four candidates each, not necessarily all four.
Any cell with a subset of the four quad digits qualifies. A cell with {1, 3} is a valid member of a quad on {1, 3, 5, 7}.
More Examples
See naked quads applied in different puzzle configurations to strengthen your pattern recognition.
Row Naked Quad
Highlighted cells show the naked quads pattern
Practice Puzzles
Apply the naked quads technique on these mini challenges. Tap a highlighted cell and enter the correct digit.
Quick Reference
- Pattern:
- Four cells in a unit collectively contain exactly four candidates
- Action:
- Eliminate those four candidates from all other cells in the unit
- Look for:
- Four cells whose combined candidate sets yield exactly four digits