Subsets

Naked and Hidden Triples

Once pairs feel natural, triples are the next rung up. They work on exactly the same logic, stretched from two cells to three — and they crack positions that pairs alone cannot.

Naked triples

A naked triple is three cells in one unit — a row, column or box — whose candidates are drawn entirely from the same three numbers. The catch that trips people up is that each cell need not hold all three. As long as every one of the three cells contains only numbers from that trio, and between them they cover all three, the set is complete. Say three cells show {2,5}, {5,8} and {2,8}: no single cell has the full {2,5,8}, yet together the three will consume the 2, the 5 and the 8 for the whole unit. That lets you strike those three numbers from every other cell in it.

Hidden triples

A hidden triple is the mirror image, wearing a disguise. Here three numbers can only go in the same three cells of a unit, but those cells also carry other candidates that make them look unremarkable. Once you notice that, say, the 1, 4 and 7 appear nowhere else in the row except those three cells, you can strip every other candidate from them — because those three cells must, between them, take the 1, the 4 and the 7. The extra numbers were noise; the hidden triple clears it away.

Why the extra cell changes things

A triple can resolve a unit that a pair leaves untouched. When candidates are spread across three cells rather than two, no pair exists to find, and a beginner concludes there is nothing to do. But the same locking logic still applies one size up: three cells that between them own three numbers seal those numbers off just as firmly as two cells owning two. Learning to see the three-cell case doubles the range of positions you can break open.

How to spot them

Naked triples fall out of a good candidate map: scan each unit for three cells whose notes, pooled together, use only three distinct numbers. Hidden triples need the opposite lens — count where each number can still go in a unit and look for three numbers sharing the same three homes. Both are easier if you have already cleared the pairs, since that thins the map and makes the triples stand out. It also helps to remember that a naked triple and a hidden triple are two views of the same fact: whenever one exists in a unit, the other is lurking in the leftover cells. Spot whichever presentation is clearer to your eye and you have effectively found both at once.

A worked naked triple

Picture a column with three cells showing {3,6}, {6,9} and {3,9}, while the column's other empty cells carry longer lists that happen to include a 3, a 6 or a 9. You cannot yet say which cell takes which digit, and you do not need to. Between them, those three cells will use up the 3, the 6 and the 9 for the entire column. So every other cell in the column can safely lose all three of those candidates. Often that trims a neighbouring cell down to a single survivor, and a placement drops out at once — a chain reaction from three cells that individually decided nothing.

Where they pay off, and where they do not

You will not need triples on easy or most medium boards; singles and pairs carry those. They earn their place on hard grids, where the board deliberately hides its structure across three cells instead of two. Do not go hunting for them prematurely — exhaust the simpler techniques first, and reach for triples only when the board genuinely stalls. Put them to work on Medium Sudoku as they start to appear, then rely on them properly on Expert Sudoku. The classic grid is always waiting at Sudoku AZ.