Naked and Hidden Triples

Naked and hidden triples are the three-cell version of the pair patterns, and they clear candidates that pairs cannot reach. A triple is three cells in one unit — a row, column or box — that together own exactly three digits. Once the pattern is confirmed, those three digits live in those three cells and nowhere else in the unit.

What the pattern is

A naked triple is three cells whose combined candidate lists contain only three digits. Each cell needs at least two of them; none of them has to hold all three. So {2,7}, {2,5,7} and {5,7} form a naked triple on {2,5,7}. However the digits settle, those three cells consume 2, 5 and 7, so every other cell in the unit loses all three.

A hidden triple is the same fact seen from the other side. Three digits appear in only three cells of a unit, even though those cells may carry a pile of other candidates. Three digits need three cells, and those three cells are all they have, so every other candidate in them is dead.

Every naked triple has a matching hidden triple somewhere in the same unit, so you never need both. Use whichever one your eyes find first.

When to look for it

• After pairs stop producing anything and the unit still has five or six unsolved cells.
• When three short candidate lists in the same unit keep repeating the same digits.
• When a digit-by-digit scan of a unit shows two or three digits crowding into the same three cells.

Triples are cheap to check and easy to miss, so scan for them before reaching for anything bigger.

How to apply it, step by step

1. Pick one unit with at least four unsolved cells and full pencil marks.
2. For the naked form, list the cells with two or three candidates and try every set of three. Take the union of their candidates.
3. If that union has exactly three digits, you have a naked triple. Delete those three digits from all other cells in the unit.
4. For the hidden form, count how many cells each missing digit can go in. Collect the digits that fit in two or three cells.
5. If three of those digits are confined to the same three cells, you have a hidden triple. Delete every other candidate from those three cells.
6. Re-check the unit and the crossing units — a triple often exposes a single immediately.

A worked example

Row 5, with candidates written under each column:

r5:  c1 {2,7}   c2 {2,4,8}  c3 {3,4,9}
     c4 {2,5,7} c5 {1,5,6}  c6 {4,8,9}
     c7 {3,6}   c8 {5,7}    c9 {1,3,4}

Look at c1, c4 and c8: {2,7}, {2,5,7}, {5,7}. Their union is exactly {2,5,7} — three cells, three digits, so it is a naked triple. Digits 2, 5 and 7 are now used up inside those three cells.

Erase them everywhere else in row 5: c2 drops the 2 and becomes {4,8}, c5 drops the 5 and becomes {1,6}. The remaining six cells now share exactly the six leftover digits {1,3,4,6,8,9}, which is the check that the deduction is sound.

Common mistakes

• Demanding three candidates per cell. Any mix of two-candidate and three-candidate cells works, as long as the union is three digits.
• Counting four digits. If the union is {2,5,7,9}, you have a quad, not a triple, and the elimination rule is different.
• Deleting the wrong side. A naked triple clears the rest of the unit; a hidden triple clears the triple cells themselves. Mixing them up destroys the grid.
• Forgetting the box. A triple sitting in a row may also sit inside one box. Sweep both units before moving on.
• Stale pencil marks. Triples depend on accurate candidates, so update them after every placement.

Where you meet it

Triples are the last of the basic techniques and the point where careful pencil marks start to pay. You will need them regularly on medium classic Sudoku and on almost every Hard puzzle, where a single triple is often the only move that reopens a stalled grid.