Two cells, two digits
Suppose two cells in a column each have the notes 2 and 3 and nothing else. One of them will be the 2 and the other the 3. Whichever way round it turns out, the column’s 2 and its 3 are both used up by those two cells.
So no other cell in that column can be a 2 or a 3, and you can erase those notes from the rest of the column. This is a naked pair: “naked” because the two digits are the only notes in their cells.
A worked example
In this position, column 8 has five empty cells. The cells in rows 2 and 6 both have the notes 2 and 3 only. They are a naked pair.
Two other cells in the column have a 2 or a 3 among their notes. In row 7 the notes are 1, 2 and 3. Take away the 2 and the 3, and only 1 is left: the pair has produced a naked single. In row 9 the notes 1, 3, 4 and 9 become 1, 4 and 9.
The pair itself is still unsolved. It did its work without being solved.
Naked triples
The same reasoning works for three cells and three digits. If three cells in a unit hold only the digits 4, 7 and 8 between them, those three digits must fill those three cells, and the rest of the unit can drop them.
The catch is that the three cells don’t each need all three digits. In this example the cells in row 9 have the notes 4 and 8, 4 and 7, and 7 and 8. No cell has more than two notes, but together they use only three digits, so they form a naked triple.
Three other cells in the row lose notes: 1, 4, 9 becomes 1, 9. Then 1, 2, 7, 8 becomes 1, 2. And 2, 8, 9 becomes 2, 9.
A naked quad, four cells and four digits, follows the same logic. It is rare, and when it appears there is usually a hidden pair in the same unit that is easier to see.
How to spot them
- Fill in notes for one unit completely before judging it. A pair found from partial notes may not be a pair.
- Look for cells with exactly two notes, then for another cell in the same row, column or box with the same two.
- Check all three units. Two cells in the same box and the same row form a pair in both, and can clear notes from both.
- For triples, look for three cells with two or three notes each and count the different digits. If the count is three, it is a triple.
Common mistakes
- Pairing cells that aren’t in the same unit. Two cells with matching notes in different rows, columns and boxes tell you nothing.
- Accepting a cell with an extra note. If one cell has 2 and 3 and the other has 2, 3 and 7, it isn’t a naked pair, though it may be a hidden pair.
- Erasing the pair’s digits from the pair itself.