Subsets

Naked and Hidden Pairs

Somewhere between easy scanning and expert chains sit the pairs. They are the techniques that get you moving again when singles dry up, and they are well worth mastering.

Naked pairs

A naked pair is two cells in the same unit — a row, column or box — that each hold exactly the same two candidates and nothing else. Say two cells both show only {4,7}. You do not know yet which is which, but between them they will use up the 4 and the 7 in that unit. That means 4 and 7 can be removed from every other cell in the unit. It is one of the cleanest eliminations in the game, and it often triggers a chain of new singles.

Hidden pairs

A hidden pair is the same relationship wearing a disguise. Here two numbers can only go in the same two cells of a unit, but those cells also show other candidates that make them look ordinary. Once you spot that, say, the 2 and the 9 appear nowhere else in the row except those two cells, you can strip every other candidate from them — because those two cells must take the 2 and the 9. Hidden pairs are harder to see than naked ones, but they unlock positions that nothing else will.

How to find them

Naked pairs jump out of a good candidate map — scan each unit for two cells with an identical two-number note. Hidden pairs need the opposite lens: count where each number can still go in a unit, and watch for two numbers that share exactly the same two homes. Both extend to triples (three cells, three candidates), but pairs alone will carry you through most medium and many hard boards.

A worked naked pair

Suppose a single row has two cells whose only candidates are {3,8}, while the other empty cells in that row carry longer lists that happen to include a 3 or an 8. You cannot yet say which of the two cells is the 3 and which is the 8, but you do not need to. Between them, those two cells will consume both the 3 and the 8 for the entire row. That means every other cell in the row can safely lose its 3 and its 8. Often that clears a cell down to a single remaining candidate, and a placement falls out immediately. One quiet observation about two undecided cells has resolved a third.

From pairs to triples and beyond

Once pairs feel natural, the same idea scales up. Three cells in a unit that between them use only three candidates form a naked triple, and those three numbers can be removed from the rest of the unit — even if each individual cell shows just two of the three. Hidden triples work the mirror image: three numbers confined to the same three cells let you strip everything else from them. You will not need triples on most boards, but recognising that pairs are just the two-cell case of a bigger family makes the harder patterns far less intimidating when they appear.

Why pairs unlock so much

Pairs are powerful out of proportion to how simple they are, because each elimination they trigger tends to cascade. Removing a candidate can create a naked single, which when placed clears more candidates, which can expose another pair, and so on. Learning to scan for two cells with identical notes — or two numbers with identical homes — is therefore one of the highest-value habits in the whole game. It is the bridge that carries you from puzzles you finish by luck to puzzles you finish by design.

Try them where they matter most on Medium Sudoku, then take the same eye to Expert Sudoku. The classic board is always ready at Sudoku AZ.