Naked and Hidden Pairs, Triples and Quads
Once scanning stalls, Sudoku becomes a game of elimination — and subsets are where that game starts. A subset is a group of cells in one unit that, between them, have a claim on the same group of digits. Two cells and two digits make a pair; three cells and three digits make a triple; four make a quad. This guide covers all of them, naked and hidden, each with a worked diagram.
Every subset technique needs pencil marks — the small candidate numbers you write into empty cells. Fill them in first, then look for the patterns below. None of these techniques place a digit on their own. What they do is clear candidates away until a hidden single or a last remaining cell appears somewhere nearby, and that's what actually fills the board.
The naked/hidden distinction runs through all of them, and it's worth fixing in your head before you start. A naked subset is visible in the cells: those cells contain nothing but the subset's digits, and you act on the rest of the unit. A hidden subset is buried among other candidates, and you act on the subset cells themselves. Same logic, opposite target.
Naked pairs
A naked pair is two cells in the same row, column, or box that contain exactly the same two candidates — and nothing else. Because those two digits must occupy those two cells between them, they can be eliminated from every other cell in the unit. It's the first real candidate-elimination technique, and the gateway to solving hard puzzles.
- the unit in focus
- the pattern
- a placement
- an elimination
In this box, two cells both hold exactly {3, 7} (amber). We don't know yet which is the 3 and which is the 7 — but we know that between them they will use up both digits. That means no other cell in the box can be a 3 or a 7. So those candidates are struck from every other cell in the box (red). In the cell that had {3, 4}, the 3 disappears and it becomes a plain 4 — a placement, handed to you for free.
The pair doesn't have to be solved to be useful. Its power comes entirely from the fact that two cells are reserved for two digits, locking those digits out of the rest of the unit.
What counts as a naked pair
Three conditions must all hold. The two cells must share the same unit — a row, a column, or a box. They must contain exactly two candidates each, no more. And those two candidates must be identical in both cells. {3, 7} and {3, 7} qualify; {3, 7} and {3, 8} do not, and neither does {3, 7} paired with {3, 7, 9}.
If the two cells happen to share more than one unit — say they're in the same row and the same box — then you can eliminate the pair's digits from both units at once. Those overlap cases are where naked pairs cascade fastest.
Finding them at the table
Scan for cells that have exactly two pencil marks — bivalue cells. Whenever you find one, glance along its row, down its column, and around its box for a twin with the identical pair. Pairs are easiest to catch right after you finish pencil-marking, while the candidate counts are fresh in your mind.
Hidden pairs
A hidden pair is two digits that can only appear in the same two cells of a unit — even though those cells also carry other candidates. When you find one, every other candidate can be wiped from those two cells, leaving just the pair. It's the trickiest of the pair techniques to spot, and one of the most satisfying.
Where a naked pair announces itself (two cells, two candidates, identical), a hidden pair is camouflaged. The two cells might show three, four, or five candidates each. The pair is "hidden" inside that clutter, and you only find it by tracking where two specific digits are allowed to go.
- the unit in focus
- the pattern
- a placement
- an elimination
Scan this column for the digits 1 and 3. Both of them can only go in the same two cells (amber) — nowhere else in the column accepts a 1 or a 3. That's the hidden pair. Now reason it through: those two cells must hold the 1 and the 3 between them, so they can't hold anything else. Every other candidate in them — here an 8 in each — is removed (red). The two cells collapse to a clean {1, 3}, which is now a naked pair as well.
Notice the direction of the elimination. A naked pair clears candidates from the rest of the unit. A hidden pair clears candidates from the pair cells themselves. Same family of logic, opposite target.
How to find a hidden pair
Go through a unit two digits at a time, or watch for digits that are scarce. For each pair of digits, count the cells in the unit where each one can go. If two different digits are both restricted to the same two cells, you've found a hidden pair — regardless of what other candidates those cells contain.
In practice the shortcut is to look for digits that appear as candidates only twice in a unit. If two such digits share the same two cells, the pattern is there. It takes more searching than a naked pair, which is exactly why beginners miss it.
Why it's worth the effort
Hidden pairs often appear on hard boards precisely where no other move is available. Because the elimination cleans up the pair cells, it frequently turns one of them into a bivalue cell that feeds a naked pair, a pointing pair, or a chain elsewhere. One hidden pair can unstick an entire region.
Naked triples and quads
A naked triple is three cells in a unit that, between them, use only three candidate digits. As with naked pairs, those three digits are reserved for those three cells, so they can be eliminated everywhere else in the unit. Quads extend the same idea to four cells and four digits. These techniques clear candidates in bulk and often crack a stalled hard board.
The subtlety that trips people up: each of the three cells does not need to contain all three digits. They only need to draw from the same pool of three. {2, 5}, {5, 8}, and {2, 8} form a perfectly valid naked triple even though no single cell shows all of 2, 5, and 8.
- the unit in focus
- the pattern
- a placement
- an elimination
Three cells in this column (amber) hold candidates drawn entirely from {2, 5, 8}. Whatever the exact arrangement, those three cells will consume the 2, the 5, and the 8 among themselves. So none of those digits can live anywhere else in the column, and they're struck from every other cell (red). One elimination like this can remove several candidates at once and immediately expose a placement.
Recognising the pattern
You're looking for any three cells in a unit whose combined candidates total exactly three distinct digits. The valid shapes are:
- Three cells each with the same three candidates: {2,5,8}, {2,5,8}, {2,5,8}.
- A mix of pairs and triples that overlap into three digits: {2,5}, {5,8}, {2,8} or {2,5,8}, {2,5}, {5,8}.
If the three cells together use a fourth digit, it isn't a triple. Counting the union of candidates is the reliable test: three cells, three digits total.
Naked quads
A naked quad is the same logic with four cells and four shared candidates. It's genuinely rare in everyday puzzles and tedious to scan for, so most solvers only reach for it on the hardest grids when nothing simpler is available. If you've mastered triples, you already understand quads — just add one cell and one digit.
Hidden triples
The mirror applies here too. A hidden triple is three digits confined to the same three cells, buried among other candidates — and as with the hidden pair, you clear everything else out of those three cells. It's rarer and harder to see than a hidden pair, and in practice most solvers find the equivalent naked subset first, after other eliminations have thinned the cells down. The logic is identical: find the digits that have nowhere else to go.
Which subset to look for first
Order matters, because each technique makes the next easier to see. A workable sweep:
- Naked pairs. Fastest to spot — you're just looking for twin bivalue cells.
- Hidden pairs. Look for digits that appear only twice in a unit.
- Naked triples. Easiest right after a round of pair eliminations, because those reduce candidate counts and make the three-cell groupings stand out. Look in units that are about half-solved.
- Quads and hidden triples. Last resort, on the hardest boards only.
There's also a counting shortcut worth knowing. In a unit with n unsolved cells, a hidden subset of size k is always accompanied by a naked subset of size n − k, and vice versa. On a nearly-full unit the naked version is smaller and easier to see; on a wide-open unit the hidden version is. Look for whichever is smaller and you'll do less work for the same eliminations.
Where subsets lead
Subsets work inside a single unit. The other family of intermediate techniques works between units, where a box and a line overlap — that's pointing pairs and box/line reduction, and the two families feed each other constantly on hard boards. For the full picture of how all eight techniques fit together, head back to the Sudoku strategies guide.
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