Hidden pairs
A hidden pair is two digits that fit in only the same two cells of a row, column or box. Those two cells must hold those two digits, so every other candidate can be removed from them. The pair is hidden because the cells usually carry extra marks that disguise it.
Hidden pairs appear from Medium puzzles upward on this site.
The mirror image of a naked pair
A naked pair is visible on the cells: two cells, the same two marks. A hidden pair is visible only when you look at the digits. In some group, the digit 3 appears as a candidate in just two cells, and the digit 9 appears in the very same two cells. Each of the group's 3 and 9 needs a home, and the only homes available are those two cells. So one takes the 3 and the other the 9. Whatever else those cells listed, 1, 4 or 7 for instance, can go.
The diagram above is a real position from this site's Medium pool. The caption states it directly: in column 1, the digits 3 and 9 fit only in r7c1 and r9c1, so every other candidate can be removed from those cells. Notation is row then column, counted from the top left.
How to find them
- Write candidates in the grid, or switch on auto-notes.
- Pick a row, column or box that still has many empty cells.
- For each digit, count the cells in the group where it appears as a candidate. Make a list of the digits that appear in exactly two cells.
- If two digits on that list share the same two cells, you have a hidden pair.
- Strip all other candidates out of those two cells.
Step three is the heart of it. A digit that appears in exactly two cells is a "locked" digit for that group, and a hidden pair is two such digits locked into the same pair of cells. If a digit appears in exactly one cell, you have a hidden single instead, which is simpler.
Why it works
Each group needs one of each digit. If 3 and 9 can only be placed in two particular cells, then those two cells must contain 3 and 9, one each. If one of those cells held any other digit, say 4, then 3 and 9 would be left competing for a single cell, and one of them would have no place in the group. That breaks the rule, so the extra candidates cannot be right. The deduction is a pure counting argument: two digits, two cells, no room for anything else.
Why hidden pairs are harder to see than naked pairs
A naked pair advertises itself because the two cells have identical, short candidate lists. A hidden pair shows up as cells with long, different lists that happen to share two digits no other cell in the group wants. Human eyes catch the first pattern in a glance and the second only by counting. That is why this site's engine, which does the counting, finds hidden pairs that a player might miss, and why the technique is graded alongside naked pairs rather than below them.
Common mistakes
- Stopping at three cells. If a digit appears in three cells, it is not part of a hidden pair. Both digits must be confined to the same two cells.
- Removing the pair's own digits. You erase the other candidates from the two cells, not the 3 and the 9.
- Missing a hidden pair that includes a hidden single. If one of the two digits appears in only one cell, it is a single and takes the cell directly.
- Applying it to the wrong group. The digits must be confined within the same group. Two digits that each have two cells in different rows do not form a pair.
Where the site's grading needs it
Hidden pairs are second-tier, alongside naked pairs, pointing pairs and box/line reduction. In the committed frequency data over 1,000 stored classic puzzles per tier, a hidden pair was used in 0 Easy puzzles, 114 Medium, 367 Hard, 251 Expert and 403 Evil puzzles. Each puzzle is solved cheapest technique first and a technique counts if used at least once, so a lower count at a higher tier simply means a cheaper technique got there first in that sample. See how difficulty is graded and the frequency guide.
Hidden pairs and bigger sets
The same idea scales. Three digits confined to three cells form a hidden triple; four to four, a hidden quad. Those are rare and hard to see, and the site's engine does not implement them. In practice the set idea is almost always faster to use in its naked form once a group has been reduced. Note too that a hidden pair in a group of nine cells is equivalent to a naked set of the remaining seven cells; finding the smaller set is easier.
In the variants
More groups mean more places for a hidden pair to hide. On the hyper board, the four shaded windows are extra groups to count in. On the irregular board, jigsaw regions are counted like boxes. On the killer board the cages do not change the digit-placement rule but they narrow candidates, which makes pairs show up earlier.
Check your understanding
- In a box, the digit 2 can go only in cells A and B, and the digit 8 can go only in cells A and B. Cell A also lists 1, 4 and 6. What is left in A?
- In a row, 5 fits in cells A and B, and 7 fits in cells A, B and C. Is there a hidden pair?
- How is a hidden pair different from a naked pair in what you must count?
The answers: A keeps just 2 and 8; no, because 7 is not confined to two cells; and for a naked pair you count candidates per cell, while for a hidden pair you count cells per digit.
A worked example
Read the diagram in the order the engine reasons. Column 1 has several empty cells. Counting where 3 can go in the column gives two cells, r7c1 and r9c1. Counting where 9 can go gives the same two cells. Nothing else in the column wants a 3 or a 9 as an option. Therefore r7c1 and r9c1 hold the 3 and the 9. At this point the two cells may still list other digits from the cells' row and box context, and those are the candidates that can be deleted. After deleting, each of the two cells has exactly two candidates, 3 and 9, and the pair has become a naked pair. Hidden pairs are often a stepping stone: once the clutter is cleared, the cells reveal their pair.
Testing yourself on a candidate grid
If you doubt a suspected hidden pair, test it by contradiction. Assume one of the two cells holds some other digit, say 4. Then 3 and 9 must both fit into the single remaining cell, which is impossible. The assumption is false for both cells, so both extras go. Running that thought experiment once or twice makes the rule feel natural, and it protects against the most frequent error, which is counting the cells wrongly and finding a pair that is not there.
Practice
Try a Medium or Hard puzzle on the main board. When you have run out of singles, fill the notes, pick a group with plenty of empty cells, and count where each digit can go. If you are not sure, the level-one hint points at the group and level two names the technique.
Previous: naked pairs. Next: naked triples. Back to the technique list.