Once you can find naked singles and hidden singles reliably, most easy puzzles fall quickly. Then you open a medium or hard grid and get stuck: every empty cell still has two, three or four pencil marks, and nothing seems to move. This is exactly where subsets come in: naked pairs, naked triples, hidden pairs and hidden triples. They rarely place a digit by themselves, but they clear away candidates so that singles appear again.
This guide explains why these techniques work, how naked and hidden subsets differ, how to find them on a real grid and which mistakes cost solvers the most time.
The idea behind every subset
All subset techniques rest on one simple counting argument. A row, column or 3×3 box (we will call any of them a unit) must contain each digit from 1 to 9 exactly once. So if you find N cells in a unit that, between them, can only hold N different digits, those cells will use up exactly those digits. No other cell in the unit can take any of them.
The mirror image is just as true: if N digits can only go into the same N cells of a unit, those cells are reserved for those digits, and every other candidate in those cells can be removed.
The first version is a naked subset, the second a hidden subset. Everything else is detail.
Naked pairs
A naked pair is two cells in the same unit that contain exactly the same two candidates and nothing else. Whichever way the puzzle resolves, one cell gets the first digit and the other gets the second. Both digits are therefore "taken" in that unit.
Worked example
Take a row that already contains 1, 3, 5 and 8. The five empty cells, labelled A to E, have these candidates:
| Cell | Candidates before | Candidates after |
|---|---|---|
| A | {2,7} | {2,7} |
| B | {2,7} | {2,7} |
| C | {2,4,6,7} | {4,6} |
| D | {4,6,9} | {4,6,9} |
| E | {2,6,7,9} | {6,9} |
Cells A and B both contain only {2,7}. One of them will be 2 and the other 7, so neither 2 nor 7 can appear in C, D or E. Removing them leaves C with {4,6} and E with {6,9}. D was not affected. Notice that the remaining three cells now share only 4, 6 and 9, which is the next step of the same logic.
If the two cells of a naked pair also lie in the same box (for example, both in row 4 and in the middle-left box), you may eliminate the two digits from both units. This is sometimes called a locked pair.
Naked triples, and why each cell does not need all three digits
A naked triple is three cells in one unit whose candidates, taken together, come from only three digits. This is the point many players misunderstand: the cells do not each need to contain all three digits. What matters is the union of their candidates. The combinations {1,5}, {5,8}, {1,8} form a perfectly valid naked triple, and so do {1,5,8}, {1,5}, {8,5}.
Worked example
Here is a column that already contains 2, 3, 6 and 7, with five empty cells P to T:
| Cell | Candidates before | Candidates after |
|---|---|---|
| P | {1,5} | {1,5} |
| Q | {5,8} | {5,8} |
| R | {1,8} | {1,8} |
| S | {1,4,5,9} | {4,9} |
| T | {4,8,9} | {4,9} |
P, Q and R together only use the digits 1, 5 and 8. Three cells need three different digits, and those are the only three available to them, so 1, 5 and 8 all end up in P, Q and R. They can be removed from S and T. Both S and T are now {4,9}, which is a brand-new naked pair, a typical chain reaction.
A useful sanity check: if you ever find four cells in one unit whose candidates come from only three digits, the grid already contains an error. Go back and check your pencil marks.
Hidden pairs
Hidden subsets look at the puzzle from the side of the digits instead of the cells. A hidden pair is two digits that, within one unit, appear as candidates in only the same two cells. Those cells may contain other candidates, which is why the pair is "hidden".
Worked example
A row contains 1, 5 and 8. The six empty cells A to F look like this:
| Cell | Candidates before | Candidates after |
|---|---|---|
| A | {2,3,6,9} | {3,6} |
| B | {3,4,6} | {3,6} |
| C | {2,4,9} | {2,4,9} |
| D | {2,4,7} | {2,4,7} |
| E | {7,9} | {7,9} |
| F | {2,7,9} | {2,7,9} |
Scan digit by digit. The digit 3 appears only in A and B. The digit 6 also appears only in A and B. Since 3 and 6 must both be placed somewhere in this row, and only A and B can take them, A and B must hold 3 and 6. Every other candidate in those two cells (2 and 9 in A, 4 in B) can go. After the clean-up, A and B have become an ordinary naked pair.
Hidden triples
A hidden triple is three digits confined to the same three cells of a unit. As with naked triples, not every cell must contain all three digits; the condition is only that these digits appear nowhere else in the unit.
Worked example
A box contains 3, 6 and 8. Its six empty cells, numbered 1 to 6, have these candidates:
| Cell | Candidates before | Candidates after |
|---|---|---|
| 1 | {1,2,4,9} | {1,4} |
| 2 | {4,5,7} | {4,7} |
| 3 | {1,7,9} | {1,7} |
| 4 | {2,5} | {2,5} |
| 5 | {2,5,9} | {2,5,9} |
| 6 | {5,9} | {5,9} |
Check where 1, 4 and 7 can go: 1 is in cells 1 and 3, 4 is in cells 1 and 2, 7 is in cells 2 and 3. All three digits are restricted to cells 1, 2 and 3, so those cells must contain exactly 1, 4 and 7. We remove 2 and 9 from cell 1, 5 from cell 2 and 9 from cell 3.
Now look at cells 4, 5 and 6: their candidates are {2,5}, {2,5,9} and {5,9}, a naked triple on 2, 5 and 9. That is no coincidence. In a unit with a given number of empty cells, a hidden subset always has a naked subset as its complement in the remaining cells. If one is hard to see, look for the other.
How to spot subsets on a real grid
- Keep complete pencil marks. Subsets only work if every possible candidate is written down. A missing candidate can make a false "pair" appear.
- Look for naked pairs first. Two identical two-candidate cells in one unit are the easiest pattern to see at a glance.
- For hidden subsets, count digits, not cells. Pick a unit and, for each missing digit, note how many cells it can go in. Digits with only two or three positions are your leads.
- Prefer units with few empty cells. In a unit with five empty cells, a naked pair automatically implies a hidden triple elsewhere, so you can choose whichever is easier to read.
- Return to singles after every elimination. The whole point of a subset is to unlock simpler moves.
Common mistakes
- Cells not in the same unit. Two {2,7} cells that share no row, column or box tell you nothing.
- Eliminating outside the unit. A naked pair in a row only clears that row, unless the two cells also share a box or column.
- Removing the wrong candidates in a hidden subset. In a hidden pair you clear the other digits from the two cells, not the pair digits from the rest of the unit (they are already absent).
- Counting the union wrongly. {1,5}, {5,8}, {1,9} is not a naked triple: together these cells use four digits.
- Trusting outdated pencil marks. After placing a digit, update its row, column and box before looking for subsets.
From triples to quads
The same logic extends to four cells and four digits: naked quads and hidden quads. They are legitimate but uncommon, and they are hard to see. Thanks to the complement rule, in a unit with seven or fewer empty cells a quad always has a simpler partner (a single, pair or triple) in the remaining cells, so you will almost always find the simpler pattern first. In a unit with eight empty cells, a hidden quad pairs with a naked quad, which is one reason true quads are reserved for difficult puzzles.
Difficulty raters reflect this progression. In Sudoku Explainer style ratings, naked pairs sit around 3.0, hidden triples around 4.0, and quads higher still. You can check how hard a specific puzzle is with our SE Rating calculator.
Put it into practice
Subsets click once you have used them a few times on real grids. Start with a medium Sudoku, fill in all candidates, and look for one naked pair in every unit before you guess anything. Then try the daily Sudoku and see how often a hidden pair is the move that breaks it open. Within a week, pairs and triples will feel as natural as singles.