Once singles run out, the next family of techniques all rest on one observation: if n cells in a region must hold n specific digits between them, those digits belong to those cells and nowhere else in the region.
Naked pairs
Two cells in the same row, column or box both have exactly the candidates {3,7} and nothing else. You do not know which is which, but you know the 3 and the 7 live in those two cells. Therefore no other cell in that region can be a 3 or a 7 — remove both digits from all of them.
The eliminations are the point. The pair itself gets you nothing directly; what it does is thin out other cells until a single appears.
Hidden pairs
The inverse. Two digits — say 2 and 6 — appear as candidates in only two cells of a region, but those cells also carry other candidates. Since 2 and 6 have to go somewhere and only those two cells can take them, those two cells must be the 2 and the 6. Every other candidate in them can be deleted.
Hidden pairs are harder to see because they are buried in long candidate lists. The way to find them is to count: for each digit, count how many cells in the region could hold it. Any two digits with a count of two, in the same two cells, form a hidden pair.
Naked removes outside, hidden removes inside
A naked pair deletes candidates from other cells. A hidden pair deletes candidates from the pair's own cells. Remembering which direction each one cuts saves a lot of confusion.
Triples
The same logic with three cells and three digits. The wrinkle is that the cells do not each need all three candidates. If three cells in a region hold {1,4}, {4,9} and {1,9}, that is still a naked triple on {1,4,9}: between them they must take all three, so those digits leave every other cell in the region.
This partial form is where triples are usually missed. Do not look for three identical cells; look for three cells whose candidates, combined, total exactly three distinct digits.
Quads
Four cells, four digits, same rule. Quads are rare and hard to spot, and by the time one exists there is usually an easier deduction available. They are worth knowing about mainly so you recognise the pattern when a solver mentions it.
Locked candidates — the technique that pairs with pairs
Two closely related eliminations, both worth more than their obscurity suggests.
Pointing: if every candidate for a digit inside a box lies in one row, then that digit must be in that row, so it can be removed from the rest of the row outside the box.
Claiming: if every candidate for a digit inside a row lies within one box, then that digit must be in that box, so it can be removed from the rest of the box.
Both are quick to check and both come up constantly on hard grids. Run through them whenever pairs stop producing results.
How to hunt efficiently
- Go region by region — the twenty-seven rows, columns and boxes — rather than scanning the whole grid at once.
- In each region, first look for cells with exactly two candidates. Naked pairs jump out from those.
- Then count each digit's candidate cells in the region. Counts of two or three are where hidden pairs and triples live.
- Apply the eliminations immediately, then check whether a single has appeared anywhere.
Every successful elimination changes the grid, so after each one it is worth a quick look for new singles before continuing the hunt.
Why this is usually enough
Singles, pairs, triples and locked candidates solve the overwhelming majority of published puzzles, including nearly everything labelled hard. Techniques beyond this — X-wings, colouring, chains — are real but come up far less often than their reputation suggests. Master this layer first.