Naked Triples and Quads

There is a naked triple when three cells of the same unit (row, column or box) have, between them, only three candidates. Each of those numbers goes in one of the three cells. You don't know yet which goes where, but all three already have their place and can be removed from the notes of the other cells in the unit.

It is the naked pair with one more cell and one more number. The difference that catches people out: none of the three cells needs to have all three candidates. It is enough that, putting the notes of the three together, no fourth number appears. Three cells with 2 and 5, 2 and 6, and 5 and 6 form a naked triple, even though each one has only two candidates.

An example

1. There are no singles, locked candidates or pairs in this grid. Column 2 is missing six numbers: the 1, 2, 4, 5, 6 and 7. These are the notes of the six empty cells.
2. Three of them, in rows 6, 8 and 9, only have the 2, 5 and 6: the one in row 6 has the 2 and the 5, the one in row 8 the 2 and the 6, the one in row 9 the 5 and the 6. None has all three, but the three together have no other number.
3. Three cells, three numbers: each cell gets one of them. The 2, 5 and 6 of column 2 are in these three cells and can be removed from the notes of the others. The 6 goes from the cell in row 1 and the 5 from the cell in row 5.
4. In box 1, the 6 could only go in the two cells of row 1. Without the one in column 2, it is left with a single place, in column 1: a hidden single.

Why does it work? If the 2 went outside these three cells, they would be left with only the 5 and the 6: two numbers for three cells, and one would end up with no number. The same goes for the 5 and the 6.

Naked quads

With four cells and four numbers the logic is the same: four cells of a unit that, between them, have only four candidates form a naked quad. Quads are rare and hard to see, because you have to put together the notes of four cells without losing count. There are no quintuples: a unit with five cells like that always has, in its other empty cells, a smaller hidden subset that gets to the same result.

The other side

A naked subset always leaves a hidden one in the other empty cells of the unit: the numbers left over can only go in the cells left over. In the example, the 1, 4 and 7 of column 2 only fit in rows 1, 3 and 5, a hidden triple. Both sides allow the same removals. When one of them is hard to see, it is worth looking for the other, which may be smaller.

What is not a triple

• Three cells that, between them, have four numbers. 2 and 5, 2 and 6, 5 and 7: that is four numbers for three cells, and none is locked.
• Three cells with no unit in common. As with the naked pair, all three have to be in the same row, column or box.

How to search

• Start with the cells that have two or three candidates. Choose one and look, in the same unit, for two others whose notes add no new number.
• Count the numbers, not the notes of each cell. What matters is the set of three: three cells, three numbers.
• Check whether it removes anything. The triple only lets you remove candidates if its numbers are still in the notes of other cells in the unit.
• After removing, look for singles, in the unit and in the boxes, rows and columns that the cleared cells cross.

Your turn

Find a naked triple (or, more rarely, a naked quad) and cross out the candidates that should be removed from the other cells of the unit. Tap each cell, choose the number and, at the end, press «Check».