Hidden singles
A hidden single is a digit that has only one possible cell left inside a row, a column or a box. The cell itself may still hold several candidates, but the digit has nowhere else to go in that group, so it must be placed there.
It is the second technique in the series and, together with naked singles, the whole of an Easy puzzle.
Two ways of looking at the same fact
A naked single asks, "What can this cell hold?" A hidden single flips the question: "Where can this digit go?" Every row, column and box must contain each digit exactly once, so if a digit has exactly one open cell in a group, that cell takes it. It does not matter what else is pencilled into the cell. Candidates that share the cell with the hidden digit can be erased as soon as you place it.
The diagram above is a real position from this site's Easy pool, drawn by the technique engine. Its caption reads: in row 2, 4 fits in only one cell, r2c7. Every other empty cell in row 2 is blocked for 4 by something in its own column or box, so the row's only possible home for a 4 is r2c7.
How to find them
The classic method is to pick a digit and sweep the grid, one group at a time.
- Choose a digit, say 4, and find every 4 already placed.
- Each placed 4 blocks its entire row, column and box for further 4s.
- Look at a box (or row, or column) that does not yet contain a 4. Count the open cells that are not blocked. If exactly one remains, that is a hidden single.
- Repeat for the other digits, favouring digits that already appear many times, because they block the most cells.
This is often called scanning or cross-hatching. It works well with a physical eye on the grid, and it works even better with notes: with candidates written in, look for a digit that appears as a mark in only one cell of a group. In the game, the highlight-same-digit option helps, because selecting a placed digit lights up every copy and makes the blocked lines easy to see.
Which group to check first
Boxes are usually the best starting point because a box blocks along both axes of its cells, and nine small groups are easier to hold in your head than a long row. Rows and columns come into their own late in a puzzle, when a line has only two or three empty cells left. A good order is boxes first, then rows and columns that are nearly full, and then any group where one digit is clearly under pressure.
Why it is certain
Each group must contain the digit. If the digit can fit in only one cell, the group would be missing it if that cell took anything else. The puzzle is promised to have a unique solution, so no valid completion leaves a digit out of a group. The deduction does not rely on any assumption about cells outside the group, and it never needs a guess.
Common mistakes
- Counting a cell that is really blocked. The usual slip is overlooking a copy of the digit in a perpendicular line. Before declaring a single, re-check every open cell in the group against both its row and its column.
- Counting an occupied cell as open. A cell that already holds a digit is not a candidate position. When you scan quickly, it is easy to count it by accident.
- Mistaking a hidden single for a pair. If a digit fits in exactly two cells of a group, that is a different situation, which later techniques such as hidden pairs and pointing pairs exploit. Only one cell makes a single.
- Ignoring the single-cell candidates after placement. The cell may have carried extra marks. Clear them, or the stale marks will hide the next naked single.
Where the site's grading needs it
Easy puzzles on this site need naked singles and hidden singles and nothing else. In the committed frequency data over 1,000 stored classic puzzles per tier, hidden singles were used in 979 Easy puzzles, 1,000 Medium, 1,000 Hard, 1,000 Expert and 999 Evil. The 21 Easy puzzles without one were solved entirely by naked singles. Each puzzle is solved cheapest technique first and a technique counts if used at least once, so these numbers show what the engine actually reached for, not every route a human might take. See how difficulty is graded and the frequency table for the method and its limits.
A habit that pays off later
Hidden singles train the question every advanced technique asks: "Where can this digit go in this group?" When a digit has two possible cells instead of one, that is the opening for pointing pairs and box/line reduction. When two digits share the same two cells, it is a hidden pair. Getting fast at the single-cell version makes the rest of the series feel like extensions rather than new topics.
A worked scan, step by step
Take the position behind the diagram. To confirm the hidden single in row 2 yourself, list the open cells of the row and ask of each one whether a 4 can go there. For every open cell except r2c7, a 4 already sits in that cell's column or in its box, so the cell is blocked. Only r2c7 survives. You never needed to know which other digits r2c7 might hold; it can be a cell with four or five candidates, and the answer is still a 4. This is the key difference from a naked single: the proof is about the digit's options within a group, not about the cell's options.
The same reasoning can be run inside a box or a column. If you cannot find a hidden single in rows, try the boxes for the same digit; the single often shows up in one view and not another.
How the engine finds them
For every row, column and box, the technique engine counts how many cells still allow each digit. A count of exactly one is a hidden single. Because the check is cheap, the engine runs it straight after the naked-single check, and both belong to the lowest tier. The hint system uses the same finder: a level-one hint points at the group, level two names "hidden single", and level three places the digit. If a naked single and a hidden single are both available, the engine reports the naked single first because it was listed earlier, which is why you may occasionally see a hint that is simpler than the one you were about to play.
In the variants
Extra groups create extra hidden singles. On the diagonal board, each of the two main diagonals is an extra group of nine cells that must hold every digit once, so you can look for hidden singles along them too. The hyper board adds four shaded windows, each with the same rule. On the irregular board the boxes are jigsaw shapes, so the "box" you scan is whatever region the cell lives in. The scanning method does not change; only the list of groups does.
Check your understanding
- A box has three open cells and needs a 6. A 6 already sits in the row of the first cell and another in the column of the second. What can you conclude about the third cell?
- Why can a hidden single sit in a cell that has five candidates?
- Which view, naked or hidden, is more natural when you are scanning a nearly complete row?
The answers: the third cell must be 6; the proof concerns the digit's places in a group, not the cell's own list; and for a nearly complete row the hidden view is usually quicker, because the row has only a few open cells to test.
Practice
Try an Easy puzzle on the main board and look for hidden singles on purpose: select a digit, let the game highlight its copies, then check each box that lacks one. When you are unsure, ask for a level-two hint, which names the technique without making the move. The 6×6 board is a quick way to practise the scan on smaller groups.
Previous: naked singles. Next: naked pairs. Back to the technique list.