Naked Singles

Every empty cell answers to three landlords — its row, its column, and its box. A digit that already appears in any of those three is off the table for this cell. Cross off enough of them and you can be left with a single survivor. That survivor is a naked single: the cell has one candidate and no choice about it.

The word naked is doing real work here. The candidate is exposed. You are not reasoning about the rest of the unit or weighing several possibilities against one another — you are looking at one cell and finding that eight of the nine digits have already been spoken for. What's left must be the answer. There is no cleverness to resist and nothing to second-guess.

It helps to hold the contrast with its cousin, the hidden single. A hidden single is a digit that has only one legal home left in a unit, even though that cell might still list several candidates on paper. There you scan a whole row or box asking where a particular number can live. With a naked single you do the opposite: you fix your attention on one cell and ask what it can still hold. Same puzzle, opposite direction. Keep your focus here on the naked single — one cell, one survivor.

Don't sweep the grid at random. Naked singles gather where the pressure is highest, so hunt in the crowded places. A cell sitting at the intersection of a nearly full row, a nearly full column, and a nearly full box has three sources of elimination bearing down on it at once, and those are the cells most likely to be pinned to a single value.

The manual method is honest and quick. Pick a promising empty cell. Read its row and note every digit already there. Read its column and add to the list. Read its box and add the rest. Now count what's missing from 1 to 9. If one digit remains, you have a naked single and you can write it in. If two or more remain, move on — this cell isn't ready, and forcing it is guesswork.

If you keep pencil marks, the work is already done for you: a naked single is simply any cell showing a single candidate. This is why it pays to update your marks as you place digits. Every number you write removes a candidate from its neighbours, and each removal can strip another cell down to one. One placement often uncovers the next, and the next, in a tidy little cascade. When a puzzle feels easy, this cascade is usually the reason.

Look at the grid above. The cell outlined for you is r1c3, and it has been left empty while much of its surroundings has filled in. To find its value, work through its three landlords in turn and keep a running tally of what's still allowed.

Start with row 1. Every digit already placed along that row is struck through in red — each one is a candidate this cell can no longer hold. Now do the same for column 3, reading down the grid and crossing off every digit that appears. Finally take box 1, the top-left block, and remove anything present there too. Between the three units, the digits 1, 2, 3, 4, 5, 7, 8, and 9 have all been accounted for.

That leaves one digit standing: 6. It appears nowhere in row 1, nowhere in column 3, and nowhere in box 1, so it is the only value that can legally sit in r1c3. The cell is shaded gold to mark it as the placement. You write 6, and you are done with it — no alternatives to test, no branch to explore. The eliminations did the deciding for you.

Notice what you did not have to do. You never asked where else 6 might go, and you never compared this cell against its neighbours. You looked at one cell, listed what its units forbade, and read off the remainder. That's the whole technique.

A naked single is not a guess dressed up as a deduction — it is forced by the rules themselves. Sudoku demands that every row, column, and box contain the digits 1 to 9 exactly once. If eight of those nine digits already appear among a cell's row, column, and box, then those eight are permanently barred from it. The cell must still hold some digit, and only one remains eligible, so that digit is the answer with certainty.

This is why you can place a naked single the instant you find one, with no need to look further ahead. Contrast that with a cell holding two candidates. There you cannot commit to either without more information, because both are still legal given what you can see. The gap between two candidates and one is the gap between a guess and a fact, and naked singles live firmly on the side of fact.

The only way to get a naked single wrong is to miscount — to overlook a digit hiding in the far end of a column, or to forget the box while checking the row. So the discipline is simply thoroughness: read all three units every time, not two. When you've genuinely covered the row, the column, and the box and found a lone survivor, that survivor is right. You can lean your whole weight on it.

Naked singles are the backbone of every solve, so make them your first move on any new puzzle and your first move again whenever you get stuck. Sweep the fullest units, place what you can, and let each placement feed the next. On an easy puzzle you may never need anything more — most of the grid falls to this one idea, one cell at a time.

When the naked singles run dry, that is your cue to look for hidden singles, and only after those to reach for the pointed elimination techniques. It is worth understanding that nearly every advanced method exists for one purpose: to rule out candidates and, in doing so, create a naked single where there wasn't one before. You do the fancy footwork to strip a cell down to one, then you place that one. The whole hierarchy of techniques leads back to this simple act.

So practise reading a cell's three units cleanly and quickly. Train your eye to drift toward the crowded corners rather than the empty ones. Update your pencil marks as you go, because a tidy set of marks turns naked single hunting into plain reading. Master this, and you will have the foundation every harder technique is built to serve.