Naked Quad Walkthrough
A Naked Quad is four cells in a unit whose candidates come from only four digits. Those digits are locked into the four cells and can be eliminated from the rest of the unit.
Below are 3 positions from real puzzles, each captured when the solver first needs this move. Find the pattern on each before-board yourself, then read the notes and check the after-board, where removed candidates are struck out in red and solved cells shown in green.
Example 1: There is a Naked Quad in Column 7
The quad is in column 7. Its cells r1c7 {4,5,6}, r2c7 {2,5,6}, r3c7 {2,4,5,6} and r6c7 {2,6} draw on just four digits, 2, 4, 5 and 6. The other four open cells in the column each lose one candidate: 6 from r4c7, 2 from r5c7, and 5 from both r7c7 and r9c7. Column 7 has eight open cells at this point, more than the guideline below suggests for quad-hunting. Treat that guideline as a rule of thumb that tells you where quads are usually found, not where they have to be.
Example 2: There is a Naked Quad in Block 5
The quad in box 5 is r4c6 {2,3,5}, r5c6 {2,3,5,7}, r6c4 {3,7} and r6c5 {2,3}: four cells and four digits, {2,3,5,7}. The other four open cells of the box are the crowded ones, r4c4, r4c5, r5c4 and r5c5, and they lose nine candidates between them. Afterwards r4c4 and r5c4 both read {1,6,8}, and r4c5 and r5c5 both read {1,6,8,9}. Those four cells now hold only 1, 6, 8 and 9, which is a second quad mirroring the first.
Example 3: There is a Naked Quad in Block 1
Box 1's quad is r2c1, r2c2, r2c3 and r3c1, and it uses only 3, 4, 7 and 8. All three box-1 cells of row 2 belong to it, which makes it easy to misread as a row pattern. It is the box that matters, because r3c1 is the fourth cell. Nine candidates fall from r1c2, r1c3, r3c2 and r3c3. After the sweep, r1c2 and r1c3 are a matching {1,5,6} and r3c2 and r3c3 are both {2,5,6}.
Why the deduction is valid
Four cells drawing on only four digits between them must use all four, locking those digits out of the rest of the unit. The logic is identical to the pair and the triplet — n cells confined to n digits consume all n — but at size four the pattern is harder to see, since the four cells may show any mix of two, three or four candidates each.
How to spot a Naked Quad on your own board
- Only look for quads in units that are already well filled. A unit with five or fewer empty cells is where a quad can actually exist and be useful.
- Take the union of candidates across four cells; a union of exactly four digits is the pattern.
- If a unit has exactly five empty cells and four of them form a quad, the fifth cell is immediately a Naked Single — that is usually the quickest way to notice one.
Common mistakes
- Hunting for quads before exhausting pairs and triplets. Almost every quad is preceded by a cheaper deduction elsewhere on the board.
- Miscounting the union. Four cells covering five digits is not a quad, and it is an easy slip when the cells hold different subsets.
What to do next
Quads usually appear near the end of a puzzle and tend to finish the unit outright. Re-scan the unit and the boxes it crosses for the singles the eliminations have created.