Three cells that consume three digits

The short version

A naked triple is three cells in one unit whose candidate lists, taken together, contain exactly three distinct digits.

Those three digits must fill those three cells. Nothing is left over for the rest of the unit, so all three digits can be deleted from every other cell in it.

The shape

The condition is about the union, not the individual lists. This is where almost everyone goes wrong.

All three of these are valid naked triples:

All three look quite different. All three consume exactly the digits 4, 5 and 7 across three cells.

What does not work: {4,5}, {5,7}, {4,7,9}. The union is four digits across three cells. There is an escape route, so nothing is spent. Delete nothing.

Why it works

Three cells, three digits, each cell holding exactly one value.

Suppose the union is {4, 5, 7}. Whatever the three cells finally contain, the set of values must be a subset of {4, 5, 7} — because no cell can hold anything outside its own list. Three cells need three values, so the subset has exactly three elements, so the cells contain 4, 5 and 7 in some order.

A unit contains each digit at most once. All three copies are now inside the triple. Every other cell in the unit is barred from all three.

As with the pair, you never learn the assignment — and the eliminations do not depend on it.

A real board

Three cells in row 5 whose candidates combine to exactly 4, 5 and 7:

• R5C1 reads {4, 5, 7}.
• R5C2 reads {4, 5, 7}.
• R5C7 reads {4, 7}.

Union: {4, 5, 7} — three distinct digits across three cells, nothing left over.

So row 5 places 4, 5 and 7 inside those three cells, in some order. Now look at R5C3, which reads {4, 5, 6}. Its 4 and its 5 both go: R5C3 drops to 6, a single candidate.

The three lists do not all have to match. R5C7 has only two candidates and still belongs, because the union is what gets spent.

Note that the cells do not look alike. That is the whole difference from a naked pair, and the reason triples are missed: solvers compare lists for equality, find none, and move on. The question is not "are these lists the same", it is "do these lists add up to three digits".

How to scan for one

Triples are found by unit, and the search is a small combinatorics problem rather than a visual one.

1. Collect short lists. In one unit, ignore every cell with four or more candidates. A triple needs three cells whose union is three digits, so every member must have at most three.
2. Look for two-plus-one. The most common triple is two cells sharing {x,y} and a third offering {x,z} or {y,z}-style overlap. Start there.
3. Test the union. Take three candidates and count distinct digits. Exactly three means a triple. Four or more means nothing.
4. Check all three unit types. A triple in a box eliminates in the box; if the same three cells share a row, it also eliminates in the row.
5. Re-read the affected cells. Like every elimination technique, the payoff is usually a cell dropping to one candidate.

A shortcut worth having: triples often hide in triples of boxes. If a row's three boxes each have a congested cell in the same column position, the union check is cheap and frequently successful.

Naked triple vs hidden triple

Direction, again.

• Naked triple: three cells are locked to three digits → remove those digits from the rest of the unit.
• Hidden triple: three digits are locked to three cells → remove the other candidates from those three cells.

A naked triple is about cells that cannot escape. A hidden triple is about digits that cannot escape. Both consume the same three-by-three structure, and both are commonly present in the same unit for different digit sets.

Hidden triples, the inward version.

The traps

Length confusion. Requiring every cell to have exactly three candidates. The union is what matters; a {4,5} cell belongs in a triple as readily as a {4,5,7} cell.

Union of four. {4,5}, {4,7}, {5,7,9} — three cells, four distinct digits. This is one of the most common false patterns, because it looks like a perfect triangle of overlap. It is not a triple, and deleting from it will break the board.

Two cells plus a bystander. Two cells forming a pair, with a third cell nearby that shares some candidates. That is a pair plus noise. The pair still works — but only as a pair, with the pair's eliminations.

Where to use it

Naked triples appear constantly in hard puzzles and are almost free once candidates are written, because the search is a filtered comparison rather than a scan of the board.

They also generalise: four cells consuming four digits is a naked quad, and the reasoning does not change at all. Quads are rare enough that most solvers never name them, but the test — union equals cell count — is the same test.