Sudoku Pointing Pairs: A Worked Example (Intersection Removal)

Pointing pairs — also called intersection removal — is the technique that turns a stuck grid back into a solvable one. It’s the natural next step after naked and hidden singles, and it’s the workhorse of nearly every “hard” Sudoku.

This guide is a worked example. We’ll name exact cells, list exact candidate digits, and show precisely which numbers get eliminated and why. By the end you’ll be able to spot pointing pairs on your own grid.

We use standard notation: r4c5 means row 4, column 5. Boxes are numbered left-to-right, top-to-bottom, so Box 1 is rows 1–3, columns 1–3.

The idea in one sentence

If a candidate digit inside a box is confined to a single row (or single column) of that box, then that digit must land somewhere on that line inside the box — so it can be eliminated from the rest of that row (or column) outside the box.

That’s the whole technique. The skill is spotting it.

Set up the example

Suppose you’re partway through a puzzle. Focus on Box 1 (r1c1–r3c3). Five cells are already filled (shown as plain digits); four are still empty, and you’ve pencilled their candidates:

Box 1 — pencil marks (filled givens shown as single digits)
         c1          c2          c3
 r1  [  2 4 7    |    8     |   1 4 7  ]
 r2  [    3      |  1 2 7   |     5    ]
 r3  [    6      |  1 2 7   |     9    ]

The four empty cells are r1c1, r1c3, r2c2, r3c2. The four digits still missing from this box are 1, 2, 4, 7.

Now spot the pointing pair. Walk the four candidate digits and ask where can each one go? The interesting one is 4: it’s pencilled in r1c1 and r1c3, but not in r2c2 or r3c2 (some earlier elimination — say a 4 already sitting in column 2 below — knocked it out of those two cells). So inside Box 1, 4 can only land in r1c1 or r1c3 — both in row 1.

That’s the spot. It isn’t obvious from the layout: the box’s empties span three different rows, yet candidate 4 is squeezed onto just one of them. That is a pointing pair.

Make the deduction

Box 1 must contain a 4 somewhere — every box holds each digit exactly once. We just showed the only cells that can take it are r1c1 and r1c3, both on row 1. Therefore the 4 for Box 1 is guaranteed to sit on row 1.

That means no other cell in row 1 can be a 4. Look across the rest of row 1 — the cells outside Box 1, namely r1c4 through r1c9 — and strike out 4 wherever it appears as a candidate.

Say row 1 currently held 4 as a candidate in r1c5 and r1c8:

1. Eliminate 4 from r1c5.
2. Eliminate 4 from r1c8.

We changed nothing inside Box 1 — we don’t yet know whether the 4 is in r1c1 or r1c3. But the rest of row 1 just lost two candidates for free.

Why the eliminations matter

Eliminations are not the goal; they’re the lever. Suppose r1c5 had candidates 7 before. Removing the 4 leaves 7 — a naked single. Place 7 in r1c5, and the chain continues: that 7 now clears 7s from row 1, column 5, and Box 2.

One pointing pair cracked open a cell that singles alone could never reach. That’s the entire reason intermediate techniques exist — they manufacture the singles that finish the puzzle.

Pointing on a column (the mirror case)

The same logic runs vertically. Picture a different grid: in Box 4 (r4c1–r6c3), candidate 2 is pencilled only in r4c1 and r6c1 — both in column 1. Then the 2 for Box 4 must be in column 1, so eliminate 2 from the other column-1 cells that still carry it as a candidate. Say 2 is also pencilled in r2c1 and r8c1 — strike it from both. (Only touch cells where 2 is genuinely still a candidate; filled cells and cells that already ruled out 2 are irrelevant.)

Row or column, the rule is identical: confined to one line inside a box → cleared from that line outside the box.

How to scan for it efficiently

You don’t need to check every box for every digit. Scan with intent:

1. After singles stall, pick a box that still has several empty cells.
2. For each unplaced digit in that box, ask: are all its candidates on one row, or one column?
3. If yes, you have a pointing pair (or triple). Follow that line out of the box and delete the digit everywhere else on it.
4. Re-check for new singles. Repeat.

Boxes that are two-thirds full are the richest hunting ground — fewer candidate cells means digits are more likely to be confined to a single line.

Pointing pairs vs. box/line reduction

These two are mirror images and easy to confuse:

• Pointing pair: candidates confined within a box point out along a line — you eliminate along the row/column.
• Box/line reduction (claiming): candidates confined within a row/column point into a box — you eliminate within the box.

Same intersection, opposite direction. Learn pointing pairs first; box/line reduction will then feel obvious.

Practice it on a real grid

The fastest way to internalize pointing pairs is to hunt for one yourself. Puzzmint’s free browser beta is live now — play a Sudoku straight away, no download and no account. A fresh daily pair of logic puzzles lands at launch; for now the beta is open whenever you want to practise.

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For the bigger picture, see the complete how-to-solve-sudoku guide and the intermediate techniques roundup, which puts pointing pairs alongside hidden pairs and X-wings.

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