Sudoku Intermediate Techniques: Naked Pairs, Pointing Pairs & Box-Line Reduction
You have solved easy puzzles. You know how to spot a naked single and run a hidden-single check through every box. Now a medium-rated puzzle shows up, and the grid goes quiet — no more obvious moves.
That is not a dead end. It means you have reached the point where pencilmarks and intermediate techniques take over.
Three techniques unlock medium and most hard-rated puzzles: Naked Pairs, Pointing Pairs, and Box-Line Reduction. Each one finds a group of cells where logic forces a digit into a specific place — then eliminates it everywhere else. Together they carry you from “stuck on medium” to “finishing hard puzzles confidently.”
Pencilmarks Are Not Optional Here
Before you can use any intermediate technique, you need pencilmarks (also called candidates) filled in across the whole grid.
A pencilmark is a small digit written in a cell’s corner to track which values are still possible there. Most digital apps have a pencilmark mode you can toggle. On paper, write them small — one digit per candidate.
Here is the approach:
- Scan the grid and fill any naked or hidden singles you can find right away.
- After no more singles remain, go cell by cell and write every possible digit (1–9) that is not already eliminated by the cell’s row, column, or box.
- Now you have a complete candidate picture — and intermediate techniques become visible.
Every technique below works by looking at pencilmarks and asking: “given what I see here, which candidates can I definitively remove?”
Technique 1: Naked Pairs
You have already used naked singles — a cell with only one candidate left. A naked pair extends that idea across two cells.
A naked pair is two cells in the same row, column, or box that both contain exactly the same two candidates — and nothing else.
Here is why that matters. Those two digits have to go in those two cells. We do not know yet which goes where, but we know for certain that no other cell in the shared group can hold either of those digits. So we can remove both candidates from every other cell in the row, column, or box they share.
Worked example — Naked Pair in a row:
Look at Row 3. After filling pencilmarks, you notice:
- R3C2 has candidates
{3, 8}only. - R3C7 has candidates
{3, 8}only.
Both cells share row 3. Both hold exactly the same two candidates. This is a naked pair.
Conclusion: digit 3 and digit 8 will land in R3C2 and R3C7 — in some order. That means every other cell in row 3 cannot hold a 3 or an 8. Erase 3 and 8 from the pencilmarks of R3C1, R3C3, R3C4, R3C5, R3C6, R3C8, and R3C9.
After that erasure, some of those cells may have only one candidate left — becoming instant naked singles. One naked pair can cascade into several free placements.
Naked triples work the same way but with three cells. The three cells in the same group collectively span at most three digits — each cell may hold two or three of those candidates. Remove all three digits from every other cell in the shared group.
The search pattern: After filling all pencilmarks, scan each row, column, and box for any two cells that both have exactly two candidates. Check whether they share the same pair. That is your naked pair.
Technique 2: Pointing Pairs
Pointing Pairs work differently. Instead of looking at which cells share the same candidates, you look at where a single digit can go within a box — and notice that all its options line up in one row or column.
Here is the setup: pick a box and pick a digit. Check every empty cell in the box. Sometimes, pencilmarks show that the digit can only go in two or three cells — and all of those cells happen to sit in the same row or column.
When that happens, the digit is locked into that row or column for its placement in this box. It will not land anywhere else in the row or column outside the box. So you can safely remove it from every other cell in that row or column.
Worked example — Pointing Pair on digit 3:
Focus on the top-right box (rows 1–3, columns 7–9). R1C8 already has a digit placed, so it is not a candidate for 3. That leaves digit 3 as a candidate in exactly two cells: R1C7 and R1C9 — both in row 1.
Since digit 3 must go somewhere in this box, and its only two options are both in row 1, digit 3 is guaranteed to land in row 1. That means no other cell in row 1 can hold a 3.
Remove 3 from R1C1, R1C2, R1C3, R1C4, R1C5, and R1C6 — every cell in row 1 that sits outside the top-right box.
The search pattern: For each digit 1–9, go box by box. Find all cells in the box where that digit is still a candidate. If they all share a row or column, you have a pointing pair (or triple). Remove the digit from the rest of that row or column.
Technique 3: Box-Line Reduction
Box-Line Reduction is the reverse of Pointing Pairs. Instead of starting inside a box, you start inside a row or column and notice that all candidates for a digit line up inside one box.
The logic is a mirror image: if a digit can only go in one box along a given row or column, then that digit is guaranteed to land in that box. It will not appear anywhere else in the box from a different row. So you can remove it from every other cell in the box that is outside the row or column.
Worked example — Box-Line Reduction on digit 8:
Look at row 6. Pencilmarks show digit 8 as a candidate in exactly two cells in row 6: R6C4 and R6C6. Both cells fall inside the center box (rows 4–6, columns 4–6).
Since digit 8 must go somewhere in row 6, and row 6’s only 8-candidates are both inside the center box, row 6’s 8 is going to land inside the center box. That means no other cell in the center box can hold an 8 from a different row.
Remove 8 from R4C4, R4C5, R4C6, R5C4, R5C5, and R5C6 — every cell in the center box that is not in row 6.
How to scan for it: Take each digit and sweep through the grid row by row: list every cell in a given row where the digit is still a candidate. When all those cells sit inside a single box, Box-Line Reduction applies. Repeat the sweep for columns — the same pattern, just rotated 90 degrees.
Putting the Three Techniques Together
When you sit down with a medium or hard puzzle, work through this sequence after your initial scan:
- Scan first — fill any naked or hidden singles before anything else (by eye, no pencilmarks yet).
- Fill all pencilmarks once no more singles remain.
- Search for naked pairs in every row, column, and box.
- Search for pointing pairs — go box by box for each digit.
- Search for box-line reduction — go row by row and column by column for each digit.
- After each elimination, recheck for naked/hidden singles — eliminations frequently unlock them.
The order matters because each step can create new opportunities for the steps above it. After a box-line reduction clears a candidate from several cells, a naked pair may appear that was not visible before. Cycle through all three checks until nothing remains, then move up to advanced techniques if the puzzle is still unsolved.
Recap Checklist
After finishing this guide, you should be able to:
- Fill complete pencilmarks across the whole grid before applying intermediate techniques.
- Identify a naked pair: two cells in the same group with the same two candidates — and remove those candidates from the rest of the group.
- Identify a pointing pair: candidates for a digit in a box that all share one row or column — and remove that digit from the rest of that row or column.
- Identify box-line reduction: candidates for a digit in a row or column that all share one box — and remove that digit from the rest of that box.
- Re-run your naked/hidden-single check after every elimination, since new singles often appear.
- Cycle through all three intermediate techniques before giving up and reaching for advanced techniques.
Frequently Asked Questions
Q: Do I need to fill pencilmarks for every cell before I start?
Yes, for intermediate techniques. A naked pair is invisible until you can see both cells’ full candidate lists. Fill pencilmarks completely after exhausting naked and hidden singles, then apply the techniques above.
Q: What is the difference between a pointing pair and box-line reduction?
Direction. Pointing pairs start inside a box and push eliminations outward into a shared row or column. Box-line reduction starts inside a row or column and pushes eliminations inward into a shared box. They are mirror images of each other.
Q: What is a naked triple?
A naked triple is three cells (not digits) in the same group whose candidates collectively span at most three distinct digits — each cell may hold two or three of those digits. Since the three cells together account for all three digits, remove all three from every other cell in the shared group. The key: it is the count of cells (three) that defines a triple, not the number of digits any one cell holds.
Q: When should I move on to advanced techniques?
When you have applied all three intermediate techniques, re-checked for naked and hidden singles, and the puzzle still has no moves. At that point, the puzzle requires X-Wing, Swordfish, or Y-Wing.
What to Try Next
- Sudoku Series Hub — Full technique overview and level chooser.
- Advanced: X-Wing, Swordfish, Y-Wing — The next step when intermediate techniques are not enough.
- The X-Wing Technique: Deep Dive — Extra X-Wing examples and the column-based variant.
- Naked Singles and Hidden Singles: Sudoku’s Foundation — Revisit the beginner foundation with deeper coverage.
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