Naked Triples in Sudoku
A naked triple is the natural next step after naked pairs — three cells locking up three digits instead of two cells locking up two. The logic is the same, but triples come with a twist that trips up almost everyone at first: the three cells don't each need to contain all three digits. Once you internalize that, naked triples become a reliable tool for cracking hard and expert puzzles.
What Is a Naked Triple?
A naked triple occurs when three cells in the same row, column, or box collectively contain only three distinct candidates. The test is on the union: put all the candidates from the three cells together, and if the combined set has exactly three digits, you have a triple.
Those three digits must go into those three cells — one each. So the digits can be eliminated from every other cell in the shared group.
Two shapes satisfy the definition:
- The full-set form: all three cells show the same three candidates, such as {4, 6, 8}, {4, 6, 8}, {4, 6, 8}. Easy to spot — it looks like a naked pair with one more cell.
- The rotation form: the cells hold two candidates each, cycling through the set — {2, 5}, {5, 8}, {2, 8}. No cell contains all three digits, yet the union is exactly {2, 5, 8}. This is the form most players walk past.
Any mix in between (one full cell, two partial) also counts. The union is the only thing that matters.
Why Does This Work?
Count cells against digits. Three cells must eventually hold three different digits, and every candidate in those cells comes from the same three-digit set. However the digits get assigned, all three are consumed by these three cells — so no other cell in the group can hold any of them. In the rotation form the argument is identical: if a fourth cell took the 5, then {2, 5} and {5, 8} would collapse to 2 and 8, leaving {2, 8} with nothing legal to be.
How to Find Naked Triples
- Fill in complete pencil marks for the group.
- Collect the cells with two or three candidates — cells with four or more can't be part of a triple.
- Pick combinations of three such cells and take the union of their candidates.
- If the union has exactly three digits — naked triple. Eliminate those three digits from every other cell in the group.
Example 1: The Full-Set Form
In the row below, three cells are filled (9, 3, 1), leaving six empty cells for {2, 4, 5, 6, 7, 8}. C1, C4, and C7 each show exactly the same three candidates: {4, 6, 8}. Three cells, three digits — the 4, 6, and 8 are locked in, and every other cell in the row loses them:
- C3: {2, 4, 5} → remove 4 → {2, 5}
- C6: {2, 6, 7} → remove 6 → {2, 7}
- C9: {5, 7, 8} → remove 8 → {5, 7}
Example 2: The Rotation Form
Now the tricky shape. In this row, the filled cells are 4, 1, and 6, so the six empty cells share {2, 3, 5, 7, 8, 9}. Look at C1, C4, and C7: they show {2, 5}, {5, 8}, and {2, 8}. No cell holds three candidates — but the union is exactly {2, 5, 8}. Three cells, three digits: a naked triple, just wearing camouflage.
The eliminations are bigger this time, because the victims hold more of the triple's digits:
- C3: {2, 3, 5, 7} → remove 2 and 5 → {3, 7}
- C6: {5, 7, 8, 9} → remove 5 and 8 → {7, 9}
- C9: {2, 3, 8, 9} → remove 2 and 8 → {3, 9}
Notice what's left behind: the victims now read {3, 7}, {7, 9}, and {3, 9} — themselves a rotation triple of {3, 7, 9}. Every naked triple splits its group into two closed camps; seeing one camp means the other is closed too.
Naked Triples in Columns and Boxes
The logic transfers unchanged. In a column, scan the same way top to bottom; in a 3×3 box, take the union across any three of its empty cells. Box triples can be easier to spot because the cells sit close together — and when the three cells of a box triple also happen to share a row or column, the eliminations apply to both groups at once.
How to Spot Naked Triples Faster
- Start from bivalue cells. Cells with exactly two candidates are the seeds of rotation-form triples. Take a bivalue cell like {2, 5} and look for group-mates whose candidates stay inside a three-digit envelope around it.
- Scan groups with four to six empty cells. Dense groups keep candidate lists short, and short lists are what triples are made of.
- Don't demand the full set. If you only look for three identical three-candidate cells, you'll find the rare form and miss the common one. Check unions, not appearances.
Naked Quads
The same union test extends to four cells and four digits — a naked quad. Quads are rare and usually only worth hunting on expert and evil puzzles, but the search method is identical: a cluster of cells whose combined candidates form a small, closed set.
When to Use Naked Triples
Reach for triples on hard and expert puzzles, after singles and naked pairs stop producing. A single triple rarely places a digit by itself — its value is the cascade: the eliminations shrink several cells at once, and shrunken cells are where the next naked single, hidden single, or pair comes from.