Naked pairs, hidden pairs: the two techniques every solver should master

Most sudoku tutorials teach naked singles (one candidate left in a cell) and hidden singles (one cell left for a candidate in a unit). Those will solve every easy puzzle. The minute you hit medium, you need two more tools: naked pairs and hidden pairs.

These two techniques are eliminations, not placements. You don’t fill any cells with them directly. Instead, you rule out candidates elsewhere — and those eliminations cascade into placements.

Naked pair

A naked pair is two cells in the same row, column, or 3×3 box whose pencil marks have been reduced to the same two candidates.

If those two cells contain only {4, 7} as candidates, you don’t know which is the 4 and which is the 7 yet — but you do know that between them, they will use up the 4 and the 7 in that unit. So no other cell in the same unit can be a 4 or a 7.

<!-- the naked pair cells get a coral fill -->
<rect x="2" y="2" width="56" height="56" fill="#F26C5A" fill-opacity="0.25"/>
<rect x="302" y="2" width="56" height="56" fill="#F26C5A" fill-opacity="0.25"/>

<!-- pencil marks 4 7 in the two paired cells -->
<text x="20" y="22" font-size="14" font-weight="800" fill="#181614">4</text>
<text x="40" y="22" font-size="14" font-weight="800" fill="#181614">7</text>
<text x="322" y="22" font-size="14" font-weight="800" fill="#181614">4</text>
<text x="342" y="22" font-size="14" font-weight="800" fill="#181614">7</text>

<!-- other cells lose 4 and 7 -->
<text x="80" y="22" font-size="12" fill="#181614" opacity="0.85">2 9</text>
<text x="140" y="22" font-size="12" fill="#181614" opacity="0.85">5 8</text>
<text x="200" y="22" font-size="12" fill="#181614" opacity="0.85">3 5</text>
<text x="260" y="22" font-size="12" fill="#181614" opacity="0.85">2 5</text>
<text x="380" y="22" font-size="12" fill="#181614" opacity="0.85">1 8</text>
<text x="440" y="22" font-size="12" fill="#181614" opacity="0.85">3 6</text>
<text x="500" y="22" font-size="12" fill="#181614" opacity="0.85">2 8</text>

<text x="270" y="76" font-size="13" font-weight="400" text-anchor="middle" fill="#3C3732">No other cell in this row can contain 4 or 7</text>

Find a naked pair and you can scrub those two digits from every other cell in the unit. That often clears the way for a hidden single elsewhere.

Hidden pair

Hidden pairs are the same idea, run in reverse.

A hidden pair is when two specific digits can only appear in two specific cells within a unit — but those cells also have other candidates clouding the picture.

Example: in row 3, the digit 5 can only go in cells C3 or G3. The digit 8 can also only go in cells C3 or G3. The other cells in row 3 each have their own candidates, but 5 and 8 are pinned to those two.

Even though C3 might currently show pencil marks like {2, 5, 8, 9} and G3 might show {1, 5, 6, 8}, you can erase the extras. Both cells must end up as either a 5 or an 8 (since those two digits have nowhere else to go in the row). So C3 = {5, 8} and G3 = {5, 8}. You’ve just created a naked pair from a hidden pair.

<rect x="122" y="2" width="56" height="56" fill="#A7E0D3" fill-opacity="0.45"/>
<rect x="362" y="2" width="56" height="56" fill="#A7E0D3" fill-opacity="0.45"/>

<text x="14" y="22" font-size="11" fill="#181614">1 2</text>
<text x="74" y="22" font-size="11" fill="#181614">3 6 9</text>
<text x="134" y="22" font-size="11" fill="#181614">2 5 8 9</text>
<text x="194" y="22" font-size="11" fill="#181614">1 3 6 9</text>
<text x="254" y="22" font-size="11" fill="#181614">1 2 4 9</text>
<text x="314" y="22" font-size="11" fill="#181614">3 4 6 9</text>
<text x="374" y="22" font-size="11" fill="#181614">1 5 6 8</text>
<text x="434" y="22" font-size="11" fill="#181614">1 2 6 9</text>
<text x="494" y="22" font-size="11" fill="#181614">2 3 4 9</text>

<text x="150" y="50" font-size="12" font-weight="800" fill="#DC5848">5,8</text>
<text x="390" y="50" font-size="12" font-weight="800" fill="#DC5848">5,8</text>

<text x="270" y="76" font-size="13" font-weight="400" text-anchor="middle" fill="#3C3732">5 and 8 appear only in these two cells → both reduce to {5,8}</text>

Hidden pairs are harder to spot than naked pairs because the candidates are buried among other digits. The trick is to scan one digit at a time across a unit: where can the 5 go in this row? If the answer is only two cells, check the next digit. If another digit also pins to those same two cells, you have a hidden pair.

Why these two matter more than naked triples, X-wings, etc.

Naked and hidden pairs appear in roughly 80% of medium and hard puzzles. The fancier techniques you’ll read about — naked triples, X-wings, swordfish, coloring — exist for the puzzles where pairs alone don’t crack the puzzle. Most of the time, pairs are enough.

The other reason: pairs are fast. Once you know what you’re looking for, you can scan a row in 3–4 seconds. A swordfish takes 30 seconds even when you find one.

Build the habit:

1. Mark all candidates before you start looking for pairs (use the Notes mode in our puzzle — keyboard shortcut N).
2. Walk the rows, columns, and boxes one at a time.
3. For each unit, ask: Are any two cells identical to a 2-candidate set? (naked pair).
4. Then ask: Are any two digits only present in the same two cells? (hidden pair).
5. When you find one, apply the elimination and move on. Don’t try to solve a chain immediately — let the next pass pick up the cascade.

Pairs in killer sudoku specifically

Killer adds cage constraints, which interact beautifully with pairs.

If you’ve worked out that a 3-cell cage worth 6 must contain {1, 2, 3} (from the cage combinations cheat sheet), and two of those three cells share a row, the third cell pins down — and the {1, 2, 3} effectively act as a naked triple for the other six cells of that row.

Combining cage combinations + naked pairs is where killer sudoku stops feeling random and starts feeling like a logic puzzle that you can grind through deterministically. Which it is.