Hidden Singles in Sudoku
Hidden singles are the technique that separates easy solvers from medium solvers. Once you understand them, a large class of previously "impossible" cells becomes immediately solvable. This is the most important intermediate skill to master — more medium puzzles are cracked by hidden singles than by any other technique.
What Is a Hidden Single?
A hidden single occurs when a particular digit can only be placed in one cell within a row, column, or 3×3 box — even though that cell appears to have multiple possible candidates.
The word "hidden" describes the fact that the uniqueness isn't visible when you look at the cell alone. You have to look at the group (the entire row, column, or box) to see that only one cell in it can hold that digit.
Hidden Singles vs. Naked Singles
The difference is the direction of your analysis:
- Naked single: You look at one cell and ask, "What digit can go here?" Only one digit is possible.
- Hidden single: You look at one group and ask, "Where can this digit go?" Only one cell in the group can hold it.
Both produce the same result — a definite digit placement — but they require different thinking. A naked single is found by examining a cell; a hidden single is found by examining a group. On medium puzzles, hidden singles are far more common than naked singles, so developing the group-scanning habit is essential.
How to Find Hidden Singles
Pick any digit, say 6. Go through each group (all 9 rows, all 9 columns, all 9 boxes) that doesn't yet contain a 6. For each such group, look at the empty cells and ask: which ones can hold a 6?
A cell cannot hold a 6 if a 6 already exists in the cell's row, column, or box. Apply these constraints to each empty cell in the group. If only one empty cell survives — place the 6 there.
Step-by-step:
- Choose a digit to scan for (start with the most frequent one on the board).
- Find all groups (row, column, or box) that don't already contain that digit.
- Within each incomplete group, identify the empty cells.
- For each empty cell, check: does a copy of the digit exist in this cell's row? Column? Box?
- Cross off any cell where the answer is yes to any of the three checks.
- If exactly one empty cell remains uncrossed in the group — that's a hidden single.
Example: Hidden Single in a Box
Consider the center 3×3 box below. It has four empty cells and you are looking for where 3 can go. Two of the empty cells share a column with an existing 3 elsewhere on the grid — eliminated. A third empty cell shares a row with an existing 3 — eliminated. Only one empty cell remains: that cell must be 3.
From looking at that cell itself, it might appear to have multiple candidates. But from the box perspective, it is the only place 3 can go — making it a hidden single.
Example: Hidden Single in a Row
Hidden singles appear in rows and columns just as often as in boxes. The logic is identical: you scan the row for a specific digit, cross off cells blocked by column or box constraints, and look for the only surviving candidate.
In the example below, row 5 has five filled cells and four empty cells. You are scanning for the digit 7. Columns 2, 4, and 6 each already contain a 7 elsewhere on the grid — those columns block three of the four empty cells. Only the cell in column 8 remains available. It must be 7.
This is the row version of the hidden single. The scanning method is exactly the same; only the orientation of the group changes. Always check rows and columns, not just boxes.
Scanning All Three Group Types
Hidden singles can appear in rows, columns, or boxes. Always scan all three:
- Box hidden singles are usually easiest to spot visually because the nine cells are clustered together.
- Row hidden singles require scanning nine cells horizontally. Existing digits in their columns do the eliminating.
- Column hidden singles require scanning nine cells vertically. Existing digits in their rows do the eliminating.
Many solvers start with box scans because the grouped layout makes patterns obvious. Then they switch to row and column scans for the digits that didn't yield a box single. Practice all three orientations equally — puzzle designers plant hidden singles in all of them, and an expert solver doesn't skip any.
Which Digit to Scan First?
You can scan for any digit in any order, but starting with the most-placed digits saves time. If the digit 7 already appears six times on the board, then only three of the nine boxes still need it. Scanning three boxes is faster than scanning nine. More importantly, six existing 7s create six rows and six columns that are blocked for 7 — this means the remaining three placements are heavily constrained and likely to produce hidden singles immediately.
A practical routine: count how many times each digit appears on the board. Rank them from most to least frequent. Start scanning with the top two or three digits. If a scan for a particular digit doesn't find any hidden singles, move on to the next digit rather than spending extra time. Efficiency compounds — a single placement often creates new hidden singles for other digits by adding one more blocking constraint.
Hidden Singles and Pencil Marks
With pencil marks, hidden singles become more visible. Fill in all candidates for every empty cell. Then, for each digit, look through each group. If a digit appears as a pencil mark in only one cell of a group, circle it — that's your hidden single.
This is one of the main reasons experienced solvers always maintain pencil marks on medium and hard puzzles. Without them, you rely on visual cross-hatching, which becomes error-prone as the grid fills up with constraints. Pencil marks make hidden singles explicit: you're looking for a digit that appears as a candidate in exactly one cell per group.
How Many Hidden Singles Per Puzzle?
On a typical medium puzzle, the majority of placements come from hidden singles. After naked singles are exhausted — usually within the first few moves — a medium puzzle has a constant supply of hidden singles throughout. Each placement unlocks more. A well-constructed medium puzzle can be solved entirely with naked singles and hidden singles, with no need for pencil marks if you use cross-hatching systematically.
On a hard puzzle, hidden singles are less abundant and tend to dry up partway through. You'll find them at the start while the grid is sparse, then hit a wall where no hidden single is immediately available. This is the signal to switch to more powerful techniques: naked pairs, pointing pairs, or box-line reduction.
When Hidden Singles Are Exhausted
When scanning all 9 digits across all rows, columns, and boxes produces no new placements, it's time to advance. The next step is almost always to set up full pencil marks across the entire grid, then search for naked pairs or pointing pairs. These techniques require pencil marks to find reliably, and they unlock further hidden singles by reducing candidates in cells you previously couldn't determine.
Hidden singles are never truly "done" — a naked pair elimination often creates a new hidden single that wasn't visible before. Always re-scan for hidden singles after any elimination step, no matter what technique produced it.