Naked and Hidden Pairs

Naked and hidden pairs are the first technique that removes candidates instead of placing digits, and they are what gets you past the point where singles dry up. Both patterns live inside a single unit: one row, one column or one box.

What the pattern is

A naked pair is two cells of a unit holding the same two candidates and nothing else. Those two digits must occupy those two cells in some order, so no other cell in the unit can use them.

A hidden pair is two digits that are legal in only two cells of a unit. Those cells have to take the two digits, so every other candidate sitting in them is dead and can be erased.

The two are mirror images of each other. Clear the extras out of a hidden pair and what remains is a naked pair.

When to look for it

• As soon as a singles sweep stops producing placements.
• In units with four to six empty cells, where two locked digits bite hardest.
• Around bivalue cells. Any cell with exactly two candidates is a possible half of a naked pair, so check its units for a twin.
• After a big elimination elsewhere, which often trims a third cell down to the same two digits.

How to apply it, step by step

1. Choose a unit and write out its candidates.
2. Look for two cells with an identical two-candidate list. That is a naked pair.
3. Remove both digits from every other cell of that unit.
4. Now count instead: for each digit still missing, note how many cells could take it.
5. If two digits share exactly the same two cells, that is a hidden pair. Delete all other candidates from those two cells.
6. If the pair cells also share a second unit, for example a row and a box, apply the same eliminations there as well.

A worked example

Naked pair in row 7:
   c1     c2      c3    c4  c5        c6   c7      c8   c9
  {2,6}  {1,4,9} {2,6}  [5] {1,4,6,9} [3]  {1,4,7} [8]  {4,7,9}

  c1 and c3 both read {2,6}, so 2 and 6 are used up in row 7.
  Remove 6 from c5  ->  c5 becomes {1,4,9}

Hidden pair in box 1 (4 and 6 already placed):
  {1,3,5,9}  {3,5,7}  {1,7,9}
  [4]        {3,5,8}  {5,7,8}
  [6]        {2,3,8}  {2,5,8}

  1 is legal only in the two top corner cells, and so is 9.
  Both cells reduce to {1,9}: drop 3 and 5 from the left one,
  drop 7 from the right one.

Neither pair placed a digit, but the box now has two cells locked, which usually exposes a single on the next sweep.

Common mistakes

• Calling two cells a pair when one has a third candidate. Two cells reading {2,6} and {2,6,7} are not a naked pair. They may belong to a triple instead.
• Working across units. Both cells must share a row, a column or a box. Two lookalike cells in unrelated units prove nothing.
• Erasing inside a naked pair. The pair cells keep both candidates. It is the rest of the unit that loses them.
• Skipping the hidden version. Hidden pairs are invisible unless you count how many cells each missing digit can reach, so do that count deliberately.

Where you meet it

Pairs become necessary around medium classic Sudoku and stay useful at every level above it. They are also the pattern that trains the habit behind triples, quads and every locked-set technique further up the ladder: stop reading cells, start counting digits.