Naked singles: solve a cell with one candidate

A naked single is an empty cell with exactly one legal candidate after its row, column and box are checked. Place that digit, then update every peer that sees the cell; the deduction depends on a complete candidate list.

Build the candidate set

Treat candidates as the digits still allowed by all the givens and confirmed entries. Suppose the row excludes one, two and three; the column excludes four, five and six; and the box excludes seven and eight. The union excludes everything except nine. The cell must therefore be nine, even if none of the three units is nearly full.

That union is the key. Counting how many filled neighbors surround a cell is unreliable because the same digit may occur in more than one relevant unit. You need the distinct excluded digits, not the number of nearby clues.

Work through the example

Constraint	Digits removed
Row	one, two, three
Column	four, five, six
Box	seven, eight
Remaining	nine

First check that the cell is empty and belongs to the box you are scanning. Next inspect the entire column, including cells far above or below the current area. Finally read the remaining set. A single is a proof of placement, rather than a suggestion to try a digit.

Propagate the result

After placing nine, remove nine from other cells in the same row, column and box. A neighboring pair of candidates might shrink to a single. Follow that chain until it stops, then return to a broader scan.

For example, another cell in the row had candidates five and nine. Removing nine leaves five. That five may then simplify its own column. This cascade explains why a board can suddenly become easier after one apparently small move.

Avoid a false single

A pencilled cell displaying one mark is not automatically a genuine single. If you previously entered only the candidate you liked, the other legal digits are merely absent from your notes. Recompute the set from the board before relying on the mark.

Likewise, if a cell has no candidates, stop. That is not an unusually strong single. It points to an earlier incorrect placement or a bad elimination. Undoing or reviewing the earliest uncertain step is more useful than forcing another digit into the contradiction.

What this technique does not require

You do not need a unique-solution assumption, a pattern name or a comparison with every other empty cell. The proof is entirely local: all alternatives for this cell are forbidden. It also works inside a complicated puzzle whose other deductions are much harder.

Calculate a second single from overlapping exclusions

Consider a different empty cell, labelled T. Its row already contains one, four and eight. Its column contains two, four and nine. Its box contains three, six and eight. Repeated digits in this list are counted once, because exclusions form a set.

Calculation	Result
Start with every digit	one, two, three, four, five, six, seven, eight, nine
Remove the row's digits	two, three, five, six, seven, nine
Remove the column's digits	three, five, six, seven
Remove the box's digits	five, seven

T is not solved yet. The many filled neighbors leave two possibilities, not one. Now suppose another cell in T's row is independently solved as five. Remove five from T's notes: only seven survives, and T becomes a naked single.

This example separates the scan from the trigger. You can calculate a complete candidate list accurately and still have no placement. The new five is the information that changes T from unresolved to forced.

Follow the seven into its peers

Suppose U shares T's column and has candidates two and seven. After T becomes seven, U becomes two. A third cell V shares U's box but does not see T; V has candidates two, six and nine. U's new two removes two from V, leaving six and nine. Stop the chain there: V still has a choice.

That stopping point matters as much as the placements. Removing one candidate does not automatically solve the next square. Read the remaining set after each change, rather than assuming a chain will continue until the whole area is complete.

Try the calculation without a new clue

Return to T before the five was placed. If you erase seven merely because five seems convenient, you have manufactured a single. Both five and seven were legal at that stage. Restore them and search another square until an actual placement or valid elimination changes the candidate set. A naked single is the result of exhausting alternatives, not of hiding one in the notes.

Where to play

Practice finding the last candidate in Ninefold, where pencil notes let you keep track of each cell before committing a digit.