Naked singles and hidden singles

The difference in one sentence

Both placements are logically forced. The difference is where you direct your attention. For a naked single, calculate everything that can enter one square. For a hidden single, follow one digit across an entire unit and see where it can still go.

Worked example 1: a naked single

In the green central square below, check all three constraints. Its row is missing 2, 5, 6, 7 and 9. Its column allows only 3, 4 and 5. The centre box removes 3 and 4. The intersection of those lists is just 5, so row 5, column 5 must contain 5.

A fast naked-single routine

1. Choose an empty cell in a nearly complete row, column or box.
2. Write the digits missing from its row.
3. Remove any digit already present in its column.
4. Remove any digit already present in its box.
5. If one candidate remains, place it and rescan the affected units.

Worked example 2: a hidden single

Now inspect row 3 rather than one square. The row is missing 1, 2, 3, 4, 5 and 7. Candidate 5 is blocked from every empty position except column 7. The green cell still has several candidates—1, 3, 4, 5 and 7—so it is not naked. However, 5 has no other home in the row. That makes 5 a hidden single at row 3, column 7.

How to search for hidden singles reliably

Choose one unit—usually a box with several givens. Identify a missing digit, then use the crossing rows and columns to eliminate cells. This is sometimes called cross-hatching. Repeat for each missing digit before moving to the next unit.

A second approach is candidate counting. If you already have complete notes, scan all appearances of a digit within a row, column or box. One appearance means a hidden single. Two appearances may become a locked candidate or pair later.

Why singles often arrive in chains

Placing a single removes that digit as a candidate from its peers. One of those peers may now have a single candidate, and a neighbouring box may gain a hidden single. Follow the chain until it stops. This is why rescanning locally after every placement is more productive than jumping to an unrelated corner.

Common mistakes

Calling every forced number a naked single

The name matters because it records your reasoning. If a cell had several candidates but one digit had nowhere else in the unit, you used a hidden single.

Using incomplete notes as proof

A digit appearing only once in your notes is meaningful only if the notes for that unit are complete and correct. When in doubt, recheck the row, column and box from the givens.

Forgetting the box

Rows and columns are visually long, so solvers sometimes omit the 3×3 box. A naked single requires all three constraints.