Naked Pairs/Triples

This technique is known as "Naked Subset" or "Disjoint Subset" in general, and works by looking for candidates that can be removed from other cells. Naked Pairs are when there are just two candidates being looked for, Naked Triple when there are three, and Naked Quads when there are four.

Take a look at the bottom row on this puzzle, which has already been mostly completed.

Sudoku puzzle

People might describe the contents of the area in terms of either a single value or a set of candidates, with each cell's contents in curly brackets { }. The bottom row would look like this:

{1369} {15} {4} {369} {8} {7} {15} {16} {2}

You don't have to worry about the cells which already have their value set, so you can look at just the ones with several candidates:

{1369} {15} {369} {15} {16}

Again, look at the bottom row :

Sudoku puzzle

You should see the pair {15} in two places highlighted.

You don't know which of those cells is the 1, and which is the 5, but you can be sure that between them, they only contain 1 and 5.

This doesn't sound like much until you realise that if those cells contain 1 and 5, then none of the other cells in that area can contain them - so you can remove 1 and 5 as candidates from all of the other cells in the area!

Carrying on from here :

Sudoku puzzle

See that we've been able to remove two 1s as candidates from other cells? Even better, that has left a 6 as a single candidate! The puzzle is now much easier to solve!

Spotting these pairs is quite easy, but the same technique can also be applied to larger groups, of triples and quads. You might also see this technique called "Disjoint Subsets"

An example of a Naked Triple might be:

{1578} {4} {569} {569} {25} {1589} {569} {27} {3}

Can you spot that {569} occurs three times? That means that the values 5,6,9 exist in only those three cells - and can't exist in any of the other cells. After you remove the candidates from other cells you end up with:

{1578} {4} {569} {569} {25} {1589} {569} {27} {3}

Which becomes

{178} {4} {569} {569} {2} {189} {569} {27} {3}

(And you can now see the single candidate 2!)

Getting Clever

A bit trickier still is that quite often you can apply the same technique, even if it doesn't look like an obvious triple.

Look in this highlighted area:

Sudoku puzzle

There's actually a triple in there you can work with, even though it isn't complete! Look for 1s, 3s and 8s.

If you were to write it out, you'd see the following:

{149} {18} {1589} {38} {45} {7} {138} {6} {2}

The trick is to look for cells which only contain values out of those three candidates. (In this case 1,3,8)

{149} {18} {1589} {38} {45} {7} {138} {6} {2}

What you have are three cells which between them must contain 1,3,8 and no others! Because they must contain just those three values, it means you can remove them as candidates from other cells:

{149} {18}

{1589} {38} {45} {7} {138} {6} {2}


You might often see this in puzzles with three cells containing just two values each, for instance {24} {47} {27}. Again, there's just three values shared between three cells, so you can remove 2,4 and 7 from any other cells in that area!

Why are they called "Naked"?

They are called naked because whether they contain all of the set you're looking for or not, they won't be hidden underneath any other candidates. In the 1,3,8 example above, the naked triple only contains those values, and nothing else to hide behind!

Can you find the triples in these puzzles?

Sudoku puzzle

Sudoku puzzle

Sudoku puzzle

Sudoku puzzle

Sudoku puzzle

What about Quads?

Quads are much harder to spot – because each cell in the quad might have 2,3 or all 4 of the candidates for the quad. It really takes a long time to look for these by eye, and in general you’ll only find these in really tough puzzles.

Can you spot the quad for 1,3,5 and 7 in this puzzle?

Sudoku puzzle