Naked and Hidden Pairs in Sudoku (Plus Triples and Quads)

Pairs and triples are where Sudoku stops being about finding numbers and starts being about clearing pencil marks. They rarely place a digit on their own. Instead they remove a few candidates, and very often that is just enough to open up a single somewhere else. This guide explains each pattern, shows it on a real position (every diagram comes from our solver, the same one behind the hints on Sudoku.my) and lists the mistakes that trip most people up.

You will need notes for all of this. If you don’t write candidates yet, read our guide to Sudoku pencil marks first, then come back.

Pairs, triples and quads: the one idea

Every row, column and box holds each digit exactly once. So if you find a group of cells inside one unit that, between them, can only hold the same number of digits as there are cells, those digits are locked into that group. Two cells and two digits make a pair, three and three a triple, four and four a quad.

Solvers often call these patterns subsets. There are two ways to find them:

• Naked: look at the cells. Their candidates, taken together, are only N digits.
• Hidden: look at the digits. N digits can only go in the same N cells.

Same logic, two directions. Once you see that, the rest is practice.

Naked pairs

A naked pair is the easiest subset to see: two cells in one unit whose notes are exactly the same two digits, such as {4,6} and {4,6}. One cell will be 4 and the other 6; we don’t know which yet, but it doesn’t matter. No other cell in that unit can be a 4 or a 6.

Look at what happens next. Row 2, column 6 had {4,5,6}; with 4 and 6 gone it can only be a 5. Row 9, column 6 had {4,9}; without the 4 it must be a 9. One pair, two new numbers. That is typical: the pair itself places nothing, but the cells around it collapse into naked singles.

A pair can sit in a box as well as in a line. If the two cells share both a box and a line (say, two cells in the same row of one box), you get to clear the digits from both units.

Hidden pairs

Hidden pairs are the same lock, seen from the digits’ side. Pick a unit and ask, for each missing digit, “where can this go?” If two digits can only go in the same two cells, those cells must be those two digits. Any other candidate written in them is wrong and can be erased.

They are called “hidden” because the pair is buried. Row 4, column 3 shows {4,5,6,7,8} and row 5 shows {1,5,6,7,8}: five notes each, nothing that looks like a pair. Only when you scan the column for 7s and 8s do you notice that no other cell can take either. After the clean-up both cells read {7,8}, and the hidden pair has turned into a naked one.

Naked vs hidden: the difference

The only real difference is where the eliminations happen. A naked subset clears its digits from the other cells of the unit. A hidden subset clears the other digits from its own cells.

Here is a neat fact: naked and hidden subsets always come in matching sets. In the naked-pair diagram above, column 6 has six empty cells. Rows 3 and 8 are the naked pair {4,6}, so the other four cells (rows 2, 4, 5 and 9) must hold the remaining digits 2, 5, 8 and 9, which makes them a hidden quad. Both views give the same eliminations, so use whichever is smaller and easier to see. Usually that is the pair.

Naked triples

A naked triple is three cells in one unit whose candidates, all together, are just three digits. Those three digits are locked into the three cells, so they leave the rest of the unit.

Notice that only one of the three cells, column 7, shows all three digits. The others have two each. That still counts, and it is the single most important thing to know about triples (more on that in the mistakes section). Three cells, three digits: wherever the 3, the 4 and the 5 end up, they fill those cells, so no other cell in row 5 can have them.

After the eliminations, look at the rest of row 5: column 1 is {6,9}, column 5 is {1,9} and column 6 is {1,6}. That is another naked triple, made of three two-digit cells. Patterns like this often chain together.

Hidden triples exist too: three digits that fit only in the same three cells of a unit. They are rarer and harder to see, and the scanning trick is the same as for hidden pairs.

A short note on quads

Quads are four cells and four digits. The logic is identical, but they are uncommon and hard to spot by eye. Because of the matching-sets fact above, a quad in a unit with few empty cells usually has a smaller hidden or naked partner that gives the same result, so most players never need to hunt for quads directly. Puzzles on Sudoku.my never require them: our solver uses pairs and triples, and the harder Expert patterns are things like the X-Wing.

How to spot them in a real puzzle

1. Get complete notes first. Subsets only work if every candidate is written in. A missing note can make a false pair look real.
2. Hunt for two-digit cells. Cells with exactly two notes are the raw material for naked pairs and triples. Scan each unit for two identical ones.
3. Check units with few empty cells. A row with four blanks can only hide small subsets, and they jump out quickly.
4. Then switch to digits. For each unit, count where each missing digit can go. Two digits sharing the same two spots means a hidden pair.
5. Look for triples last. Pairs are more common and quicker to check. If none turn up, look for three cells whose notes stay inside three digits.
6. After every elimination, look again for singles. That is where the payoff is.

If you want a wider map of where these fit, our Sudoku strategies guide covers all ten techniques in order, from hidden singles to Swordfish.

Common mistakes

• Thinking a triple needs all three digits in every cell. It doesn’t. {3,4}, {3,4,5} and {3,5} is a triple, and so is {6,9}, {1,9}, {1,6}, where no cell has all three digits. The rule is that the combined notes of the three cells are three digits.
• Using cells from different units. All the cells of a subset must share one row, column or box. Two {4,6} cells in different rows and boxes tell you nothing.
• Removing digits from the pair cells themselves. In a naked pair you erase the pair’s digits from the other cells, never from the pair.
• Mixing up the directions. Naked: clear outside. Hidden: clear inside. If you remember one line from this article, make it that one.
• Trusting incomplete notes. If you skipped a candidate, a “pair” might really be a {4,6,9} cell. Double-check before erasing.
• Stopping after the elimination. The pair is a means to an end. Always look for the single it just created.

Pairs and triples at Hard level on Sudoku.my

On Sudoku.my, each level is defined by the techniques it needs. Easy puzzles need only singles, Medium adds locked candidates, and Hard adds naked and hidden pairs and triples. Expert may also need X-Wing, XY-Wing or Swordfish. Every puzzle has exactly one solution and can be solved by logic, without guessing, so when a Hard puzzle seems stuck, a pair or triple is almost always waiting somewhere.

A few things in the app make subsets easier to learn:

• Notes with auto-remove keep your candidates tidy as you place numbers, so pairs stay visible.
• Highlighting of the row, column, box and matching numbers helps you count where a digit can go.
• Hints explain the move. You get three per puzzle, and a hint names the technique and shows the cells involved. Miko the cat would suggest using one, reading it, then finding the next pair yourself.

A good routine: play a few Hard puzzles in a row and set yourself the goal of finding one pair or triple per puzzle without a hint. Once that feels natural, you are ready for the X-Wing and the rest of the Expert toolkit.

#Techniques#Naked pairs#Hidden pairs