Hidden Singles and Naked Singles

Both examples below come from real puzzles made by the same generator as the puzzles on this site, each with exactly one solution. Rows are numbered 1 to 9 from the top, columns 1 to 9 from the left, and boxes 1 to 9 reading like a book.

The naked single: "what can go in this square?"

A naked single is a square where eight of the nine digits are already used somewhere in its row, its column or its box. The ninth digit is the only one left, so it must go there. It is called "naked" because, once you look, the answer is plain to see.

Here is one at the start of a puzzle. Take row 8, column 9, outlined below. Its row, column and box are shaded.

• Its row already holds 1, 3 and 7.
• Its column holds 2, 5, 8 and 9.
• Its box holds 2, 3, 5, 6 and 9.

Put those together and every digit from 1 to 9 appears except 4. So row 8, column 9 is 4. Notice that no single place told us the answer. The row ruled out some digits, the column others, and the box the rest. Checking only two of the three is the most common reason people miss a naked single.

The hidden single: "where can this digit go?"

A hidden single turns the question round. Pick a digit and a row, column or box that doesn't have that digit yet. If the digit can only go in one square of that unit, it goes there. The square itself might still look as if it could hold other digits. That is why the single is "hidden": you won't see it by staring at the square.

In this puzzle there is no naked single at the start at all, but there is a hidden single. Look at box 5, shaded below. It has 5 empty squares and no 2 yet.

Now check where a 2 could go in that box:

• Row 5, column 5 is ruled out: column 5 already has a 2 (at row 9, column 5).
• Row 6, column 4 is ruled out: row 6 already has a 2 (at row 6, column 3).
• Row 6, column 5 is ruled out: column 5 already has a 2 (at row 9, column 5).
• Row 6, column 6 is ruled out: row 6 already has a 2 (at row 6, column 3).

That leaves one square, row 4, column 4, so it is the 2. If you had looked at that square on its own, you would have found that both 2 and 5 were still possible there. It is not a naked single. It only gives itself away when you ask where the 2 can go.

Why hidden singles find more

A naked single needs eight different digits around one square, which is a lot to ask early in a puzzle. A hidden single only needs the other empty squares of one unit to be blocked for one digit, and a single placed digit can block a whole row or column at once. That's why medium and harder puzzles lean on hidden singles: on our medium level, every puzzle needs at least one, while our easy level can always be finished with naked singles alone.

Hidden singles work in rows and columns as well as boxes. Boxes are usually the easiest place to spot them, because a box's squares are close together and the blocking digits sit in the rows and columns running through it.

A routine for spotting singles

When you are not sure where to look next, run through this in order:

1. Digit by digit, box by box. Take the digit that appears most often on the board. For each box that lacks it, glance along the rows and columns that run through the box. If only one square is left, place it. Then move to the next most common digit. This finds hidden singles quickly, and with practice it becomes a habit.
2. Nearly full units. Any row, column or box with only two or three gaps is worth a direct check. Often each gap has one digit that fits.
3. Follow your last move. A digit you just placed blocks its row, column and box. The next single is often right beside it, or in a box it has just blocked.
4. Crowded squares. A square whose row, column and box together hold many digits is a naked single candidate. Count what it sees.

When you can't find either kind of single, it is time for notes. Our guide to pencil marks shows how to write candidates without filling the grid with clutter, and the next technique, locked candidates, uses exactly the same "where can this digit go?" thinking.

Mistakes to watch for

• Forgetting the third direction. Every square belongs to one row, one column and one box. A naked single needs all three checked. A hidden single in a box needs the rows and the columns through it checked.
• Counting a square as blocked when it isn't. A digit blocks its own row and column, and its own box, nothing else. A 5 two boxes away diagonally does not block anything in this box.
• Placing a digit that only "probably" fits. If you can't say which digits rule out every other option, it is not a single yet. One wrong digit can sit unnoticed for many moves before something breaks.

Practise

Play a few easy puzzles and look for naked singles only. Then switch to medium, where you will need the digit-by-digit scan. The game's Hint button names the technique behind each digit it fills, so you can check which kind of single you missed. For a full puzzle solved move by move, see our beginner walkthrough.