Naked triple: three cells that lock three digits

What is a naked triple?

A naked triple is three cells in the same unit whose candidates, taken together, contain exactly three different digits. Each cell may have two or three of those digits. {2,5,6}, {2,5,6}, {2,5,6} is a triple, but so is {5,6}, {2,5}, {2,6}.

When you find one, you can remove those three digits from every other cell of the unit. As with pairs, the triple does not say which cell gets which digit, only that the three digits are used up.

Why it works

Three cells each need a digit, and between them they only have three digits to choose from. They cannot repeat a digit, since they share a unit, so together they must use all three. That fills the unit’s quota of those three digits. Any other cell in the unit holding one of them would create a duplicate, so those candidates can go.

Notice that no single cell needs all three digits. What matters is the union of the three candidate lists. That is what makes triples harder to spot than pairs: there is no identical-looking pattern to catch your eye.

How to find naked triples

Pick a unit and list the empty cells that have two or three candidates. Cells with four or more candidates cannot be part of a naked triple. Then try groups of three and write down the combined set of digits. If it has exactly three digits, you have a triple.

A practical way is to start from one bi-value cell, say {2,5}, and look for cells that add at most one new digit, such as {2,6} or {5,6}. Units with five or six empty cells are the most rewarding to check, because in a unit with only four empty cells, a triple always pairs up with a single you would have found already.

A worked example

In the example board, singles, locked candidates and pairs have nothing left to give. Look at column 6. Its empty cells are R1C6 (row 1, column 6) {1,3,4,8,9}, R2C6 {3,4,8,9}, R3C6 {1,3,4,9}, R4C6 {3,5,9}, R5C6 {5,6}, R7C6 {2,5}, R8C6 {2,6} and R9C6 {5,6,8}.

The highlighted cells R5C6 {5,6}, R7C6 {2,5} and R8C6 {2,6} contain only the digits 2, 5 and 6. None of them holds all three, yet together they form a naked triple. So 2, 5 and 6 are taken in column 6, and they can be removed from the rest of the column.

R4C6 loses its 5 and becomes {3,9}. R9C6 loses both 5 and 6, which leaves a single candidate: 8, shown in gold. R1C6, R2C6 and R3C6 contain none of 2, 5 or 6, so they are unchanged.

Common mistakes

Expecting three identical cells. Most naked triples mix two-candidate cells, like the {5,6}, {2,5}, {2,6} in the example. If you only look for matching cells, you will miss most triples.

Letting a fourth digit slip in. If one of the cells is {2,5,9}, the union becomes {2,5,6,9} and the triple is gone. Always count the combined digits, not the cells.

Mixing units. All three cells must share one row, column or box, and you only eliminate inside that shared unit. Three cells spread over a row and a box do not form a triple.

Naked triples and related techniques

A naked triple is a naked pair with one more cell and one more digit, and the same pattern continues to naked quads. Its mirror is the hidden triple, where three digits are confined to three cells. Whenever a unit holds a naked triple, the remaining empty cells hold a hidden subset of the remaining digits, so look for whichever is smaller.

The three bi-value cells {5,6}, {2,5}, {2,6} also resemble the shape behind an XY-Wing. In a triple they share one unit. In an XY-Wing they are spread across units, and the logic removes a digit from cells that see both ends. On Quietdoku, triples show up in Hard puzzles, and a hint will name the technique when it is the next step.