Naked Triple in Sudoku: when three cells work together

Naked Triple is an intermediate technique that extends the reasoning of Naked Pair.

In Naked Pair, two cells in the same unit are reserved for two candidates. In Naked Triple, three cells in the same unit are reserved for three candidates overall.

This technique may seem more difficult at first, because the three cells do not necessarily need to contain all three of the same candidates. The important point is that the set of candidates present in the three cells is made up of three numbers in total.

In this guide, we will see what a Naked Triple is, how to recognize it and which eliminations it allows you to make.

A Naked Triple is formed by three cells in the same unit - row, column or box - whose candidates collectively belong to a set of only three numbers.

For example, in the same row you have three cells with these candidates:

• cell A: {2,5};
• cell B: {2,8};
• cell C: {5,8}.

Together, the three cells contain only numbers 2, 5 and 8. This means that those three numbers must occupy those three cells, in an order that is still unknown.

As a result, 2, 5 and 8 can be eliminated from the other cells in the same row.

The difference between Naked Pair and Naked Triple is the number of cells and candidates involved.

In Naked Pair, you have two cells reserved for two numbers. In Naked Triple, you have three cells reserved for three numbers.

The triple requires more attention because it can appear in different forms. It is not necessary for each cell to contain all three candidates. It is enough that the three cells, considered together, have only those three possible numbers.

For example, these three cells can form a Naked Triple:

• {1,4};
• {1,9};
• {4,9}.

Even if no cell contains all three numbers, the overall set is {1,4,9}.

To recognize a Naked Triple, you need to look for three cells in the same unit whose combined set of candidates is made up of three numbers.

Usually the involved cells have two or three candidates each. The key point is that, when considered together, they do not introduce a fourth number.

For example, these cells form a Naked Triple:

• {3,6};
• {3,6,9};
• {6,9}.

The total set of candidates is {3,6,9}. There are three cells and three candidates overall.

These cells are reserved for 3, 6 and 9, so you can eliminate 3, 6 and 9 from the other cells in the same unit.

A Naked Triple in a row occurs when three cells in the same row are reserved for three candidates overall.

Once you identify the triple, you can eliminate those three candidates from the other cells in the row.

For example, in a row you find three cells with candidates {2,7}, {2,9} and {7,9}. These three cells form a Naked Triple with numbers 2, 7 and 9.

Every other cell in the row can no longer contain 2, 7 or 9 as candidates.

Naked Triple can also appear in a column.

In this case, you need to identify three cells in the same column that, together, contain only three candidates overall.

Once you find the triple, those three numbers must occupy those three cells. The other cells in the column can no longer contain those candidates.

The reasoning is identical to the row case, but it can be harder to see because vertical reading requires more attention. This is why it is useful to keep candidates ordered and clearly visible.

Naked Triple in 3×3 boxes is very useful in medium Sudoku.

If three cells in the same box collectively contain only three candidates, those three numbers are reserved for those three cells. You can therefore eliminate those candidates from the other cells in the box.

For example, in a box you find:

• cell A: {1,6};
• cell B: {1,6,8};
• cell C: {6,8}.

The three cells form a Naked Triple with numbers 1, 6 and 8. The other cells in the box can no longer contain 1, 6 or 8.

With a Naked Triple, you can eliminate the three candidates of the triple from the other cells in the same unit.

If the triple is in a row, you eliminate in the row. If it is in a column, you eliminate in the column. If it is in a box, you eliminate in the box.

You should not eliminate candidates from cells that do not share the unit with the triple.

As with Naked Pair, Naked Triple does not directly assign the numbers to the three cells. It tells you, however, that those three numbers are reserved for those three positions and therefore cannot be used elsewhere in the same unit.

Imagine a row with these empty cells:

• cell A: {2,4};
• cell B: {2,8};
• cell C: {4,8};
• cell D: {2,4,6,8};
• cell E: {6,9}.

Cells A, B and C form a Naked Triple with candidates 2, 4 and 8.

This means that 2, 4 and 8 must occupy A, B and C. In cell D, therefore, you can eliminate 2, 4 and 8.

Cell D is left with the only candidate 6. At this point D becomes a Naked Single.

The triple did not directly solve A, B and C, but it allowed D to be unlocked.

The first mistake is thinking that the three cells must all have the same three candidates. That is not true. The combinations can be different, as long as the total set contains three numbers.

The second mistake is including a cell with a fourth candidate. If one of the three cells introduces an extra candidate that is not part of the triple, the technique does not apply.

The third mistake is eliminating candidates outside the correct unit. A triple found in a row allows eliminations in the row, not automatically across the whole grid.

The fourth mistake is looking for Naked Triple too early. It is better to check simpler techniques first, such as Naked Single, Hidden Single, Naked Pair and Hidden Pair. If you find nothing, then it makes sense to look for triples.

Naked Triple occurs when three cells in the same row, column or box collectively contain only three candidates.

Those three numbers are reserved for those three cells and can be eliminated from the other cells in the same unit.

It is an important intermediate technique because it helps unlock situations where simpler techniques are no longer enough. To use it well, you need updated candidates and an orderly reading of the grid.