Hidden Pairs in Sudoku
Hidden pairs are one of the most valuable techniques on hard and expert sudoku — and one of the most commonly missed. Unlike naked pairs, which announce themselves as clean two-candidate cells, hidden pairs hide behind extra candidates that have no right to be there. Learning to strip that camouflage away is often exactly what a stuck puzzle needs.
What Is a Hidden Pair?
A hidden pair occurs when two digits, within a single row, column, or box, can only be placed in the same two cells — even though those two cells may contain other candidates as well.
For example: suppose that in one row, the digits 2 and 7 each fit in only two cells, and it's the same two cells for both. One of those cells shows candidates {2, 3, 7, 8} and the other shows {2, 4, 5, 7}. Because the 2 and the 7 have nowhere else to go, those two cells are reserved for them — so the 3, 8, 4, and 5 can all be eliminated, leaving a bare {2, 7} pair.
Why Does This Work?
Count the obligations. The row must contain a 2 and a 7 somewhere. If both digits fit in only those two cells, then between them, those two cells must hold the 2 and the 7 — one each. Two cells, two digits, no room left over. If one of those cells were actually a 3, then either the 2 or the 7 would be left with no legal cell in the row — a contradiction. So every candidate other than 2 and 7 in those two cells is false.
Hidden Pairs vs. Naked Pairs
The two techniques are mirror images, and the names describe what you see:
- Naked pairs are found cell-first. You spot two cells that each contain exactly the same two candidates — the pair is visible, "naked." The elimination happens in the other cells of the group.
- Hidden pairs are found digit-first. You track where a digit can go within a group and notice two digits sharing the same two homes. The pair is buried under extra candidates — "hidden." The elimination happens in the pair cells themselves.
This is why cell-first scanning alone walks straight past hidden pairs: the pair cells have three or four candidates each and look unremarkable. You have to ask a different question — not "what can this cell be?" but "where can this digit go?"
How to Find Hidden Pairs
- Fill in complete pencil marks for the group you're scanning.
- For each missing digit in that row, column, or box, list the cells where it can still go.
- Note every digit that has exactly two possible cells.
- If two such digits share the same two cells — you have a hidden pair. Delete every other candidate from those two cells.
Example: Hidden Pair in a Row
In the row below, three cells are already filled (1, 6, 9), leaving six empty cells that must hold {2, 3, 4, 5, 7, 8}. Ask the digit-first question for the 2:
- C2 and C8 can't be 2 — their columns already contain a 2
- C4 and C6 can't be 2 — the center box already contains a 2
- So in this row, the 2 fits only in C3 and C7
Now scan the 7 — and it's blocked from exactly the same four cells, by a 7 in each of those columns and boxes. The 7 also fits only in C3 and C7. Two digits, same two cells: a hidden pair {2, 7}.
C3 currently shows {2, 3, 7, 8} and C7 shows {2, 4, 5, 7}. Stripping everything except the pair digits:
- C3: {2, 3, 7, 8} → remove 3 and 8 → {2, 7}
- C7: {2, 4, 5, 7} → remove 4 and 5 → {2, 7}
Check the pencil marks in the gray cells and you can verify the deduction yourself: neither a 2 nor a 7 appears anywhere in the row outside the two amber cells.
Hidden Pairs in Columns
Columns work identically. Scan each missing digit down the column, note the digits with exactly two possible cells, and look for two digits sharing the same two cells. Column hidden pairs are just as common as row pairs — the only difference is the direction you scan.
Hidden Pairs in Boxes
In the example below, the center box contains 1, 9, 8, and 6, leaving five empty cells for {2, 3, 4, 5, 7}. Column 4 already contains both a 3 and a 7, blocking the two empty cells in that column. Row 6 also contains a 3 and a 7, blocking the box's bottom row. That rules out three of the five empty cells for both digits — the 3 and the 7 fit only in R4C5 and R5C6.
- R4C5: {2, 3, 5, 7} → remove 2 and 5 → {3, 7}
- R5C6: {3, 4, 5, 7} → remove 4 and 5 → {3, 7}
Every Hidden Pair Has a Shadow
Here is a useful piece of theory. In the row example, look at what the four gray cells hold: collectively, just {3, 4, 5, 8} — four cells sharing four digits. That's a naked quad, and eliminating its digits from the rest of the row produces exactly the same result as the hidden pair. In the box example, the three gray cells share {2, 4, 5} — a naked triple, same story.
This is not a coincidence: a hidden pair is always the complement of a naked subset among the remaining cells. But tracking two digits is far cheaper than tracking a four-digit quad — which is precisely why the digit-first scan is worth learning. You find in seconds what cell-first scanning would need a quad to see.
How to Spot Hidden Pairs Faster
- Start with the crowded digits. A digit already placed six or seven times on the board has very few legal positions left — it's the most likely to be confined to two cells somewhere.
- Scan dense groups. Rows, columns, and boxes with only four to six empty cells constrain their digits the hardest, making pairs more likely.
- Re-scan after every find. A hidden pair's eliminations shrink the candidates in two cells at once, which frequently exposes a naked single, a hidden single, or even another pair.
Hidden Triples
The same logic extends to three digits. A hidden triple occurs when three digits are confined to the same three cells within a group. The cells don't each need to allow all three digits — as long as the three digits collectively fit nowhere else, every other candidate in those three cells can be removed. Hidden triples are rarer and harder to spot, but the digit-first scan finds them the same way.
When to Use Hidden Pairs
Hidden pairs typically appear on hard, expert, and evil puzzles, after naked singles and hidden singles are exhausted. If your pencil marks are complete and no cell-first pattern is visible, switching to the digit-first scan is the natural next step — and it's the same scanning habit you'll need later for X-Wings and other advanced patterns.