Naked Triples in Sudoku: Explained with Examples

A naked triple is three cells in the same row, column, or box whose candidates, taken together, are only three digits. Those three cells must hold those three digits, so the digits can be erased from every other cell in the group. It is the three-cell version of a naked pair, and it catches people out because no single cell has to show all three digits.

The rule

Find three cells in one group. Collect every candidate they contain. If the combined set is exactly three digits, you have a naked triple. Each cell can have two or three of those digits. These all count:

• {2, 5, 8}, {2, 5, 8}, {2, 5, 8}: the obvious version.
• {2, 5, 8}, {2, 5}, {5, 8}: one full cell, two partial.
• {2, 5}, {5, 8}, {2, 8}: no cell shows all three. This is the one most people miss.

In all three cases, the three cells have to use up 2, 5, and 8 between them. Three cells, three digits, no room for anything else. So 2, 5, and 8 cannot appear anywhere else in that group.

Worked example

Column 6 has six empty cells with these candidates:

• R1C6: 2, 5
• R2C6: 1, 2, 5, 9
• R4C6: 5, 8
• R5C6: 1, 8, 9
• R7C6: 2, 8
• R9C6: 1, 3, 9

R1C6, R4C6, and R7C6 together contain only 2, 5, and 8. That is a naked triple. Remove 2, 5, and 8 from the other cells in column 6. R2C6 becomes {1, 9}. R5C6 becomes {1, 9}. That is a new naked pair on 1 and 9, which strips 1 and 9 from R9C6 and leaves 3. One pattern, a follow-up pair, and a placement.

How to find naked triples

1. List the small cells. In one group, note every cell with two or three candidates. Cells with four or more cannot be part of a naked triple.
2. Try combinations of three. For each set of three small cells, merge their candidates. If you get exactly three digits, it is a triple.
3. Eliminate from the rest of the group. Only from cells in the shared group. If the three cells also share a box, eliminate from the box too.

The cycle pattern ({a, b}, {b, c}, {a, c}) is the one to train your eye for. A quick trick: when you see two bivalue cells in a group that share one digit, check whether a third cell holds the other two.

Naked quads

The same logic works with four cells and four digits. Naked quads are rare and hard to see, and in most groups a naked quad means the remaining empty cells form a hidden set that is easier to spot. If a group has five empty cells and four of them form a naked quad, the fifth cell is a single. Look for the smaller pattern first.

Common mistakes

• Mixing groups. All three cells must be in one row, column, or box. Three cells spread over two rows prove nothing.
• Counting a four-digit union. {2, 5}, {5, 8}, {2, 9} merge to four digits. Not a triple.
• Eliminating inside the triple. The pattern removes digits from the other cells. The triple cells keep their candidates.

Next up: the mirror technique, hidden triples, or the full list of sudoku techniques in solving order.