Naked pairs, triples and quads in Sudoku

1 September 2026 · 7 min read

Once singles dry up, subsets are the next thing to reach for. They are the first technique that proves something without knowing any answer — you establish that two digits are stuck inside two cells without ever finding out which goes where, and that is enough to clear them out of everywhere else.

The counting argument

All of these work for one reason. If N cells in a unit can only hold N digits between them, those N digits are used up by those N cells. Nine cells, nine digits, no repeats — so there is nothing left over for the rest of the unit.

N = 2 is a naked pair, N = 3 a naked triple, N = 4 a naked quad. The reasoning does not change; only the bookkeeping gets heavier. Note that it does not matter which digit goes in which cell, and you never find out. That is the point: you get a conclusion without the answer.

Naked pairs

Two cells in the same unit, each with exactly the same two candidates. Below, in row 8, two cells are both down to {5, 7}:

One of them is the 5 and the other is the 7. You do not know which — and you do not need to. Row 8 gets its 5 and its 7 from those two cells, so no other cell in row 8 can be either. Strike 5 and 7 from the rest of the row:

Notice the eliminations are uneven — one cell loses both digits, the other only the one it had. That is normal. A pair does not tidy the row up symmetrically; it removes what is there to remove.

Where to look. Pairs are the easiest subset to find because you are scanning for cells with exactly two marks and matching them up. Boxes are the most productive hunting ground — nine cells in a compact square, easy to compare at a glance.

Naked triples

Three cells, three digits between them. The catch that trips people up: the cells do not each need all three candidates. {1,6}, {1,6,8} and {1,6,8} is a perfectly good triple on {1,6,8}. What matters is that the union is exactly three digits and no cell strays outside it.

Below, in box 2 (the top-middle box), three cells hold {1,6}, {1,6,8} and {1,6,8}:

Those three cells take 1, 6 and 8 between them, so every other cell in box 2 loses all three:

The valid shapes of a triple, all of which behave identically:

• {a,b} {b,c} {a,c} — the awkward one, since no cell shows all three digits.
• {a,b,c} {a,b} {b,c} — the case above.
• {a,b,c} {a,b,c} {a,b,c} — the obvious one, and the rarest.

The first shape is worth memorising, because it is genuinely hard to see. Three cells whose marks are {1,6}, {6,8} and {1,8} look unrelated until you take the union.

Naked quads, and why you can mostly ignore them

Four cells covering four digits. It works — the argument is the same — but it is rarely the best use of your time, for two reasons.

First, cost. Checking every set of four cells in a unit means 126 combinations per unit and 27 units. Second, redundancy: four cells covering four digits in a nine-cell unit leaves five cells holding the other five digits, which is a hidden quint — and in practice the same position almost always yields to something cheaper first.

Learn that quads exist so you recognise one if it lands in your lap. Do not hunt for them.

The complement rule that saves half the work

In a unit with K empty cells, a naked subset of size N always comes with a hidden subset of size K − N, and vice versa. They are the same fact stated from opposite ends.

The practical consequence: in a unit with five empty cells, hunting naked triples and hunting hidden pairs are the same search. Do whichever is easier to see — and a hidden pair is usually easier, because you are counting digit occurrences rather than comparing candidate sets. Worth knowing before you spend effort on the harder half. Hidden pairs and triples covers that side.

Common mistakes

• Requiring identical candidate sets for a triple. Pairs must match exactly; triples need only a union of three. Insisting on identical sets means missing most triples on the board.
• Forgetting a pair can be in two units at once. Two cells in the same box and the same row eliminate along both. That doubles the yield and is easy to leave on the table.
• Marking up incompletely. Subsets are only valid if the candidate lists are complete. If you have been noting “interesting” candidates and skipping others, a cell that looks like {5,7} may really be {3,5,7}, and the elimination is then simply wrong.

After any subset elimination, go back and sweep for singles. Subsets never finish a puzzle themselves; they exist to create the single that does.

Every diagram on this page is a real position from a generated puzzle, and every elimination shown is checked against that puzzle’s unique solution as part of the test suite. Want a move explained on your own board? Play a puzzle.