Candidate notes—also called pencil marks—turn an unsolved Sudoku cell into a precise list of digits that remain legal. Used correctly, they expose singles, pairs, triples, locked candidates and, eventually, advanced patterns such as X-Wing. They do not replace checking the grid: every note must satisfy the cell’s row, column and 3×3 box, and every note must be updated after a placement or elimination.
What candidate notes are and why they matter
A candidate note is a small digit written in an empty cell to record a value that could still go there. For a classic 9×9 Sudoku, a digit is a legal candidate only when it is absent from all three units that contain the cell:
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- the cell’s row,
- the cell’s column, and
- the cell’s 3×3 box.
If a row already contains 4, a column contains 7, and the box contains 9, none of those digits belongs in the cell’s candidate list. Notes are working memory: they preserve the results of your checks so that you can compare cells and units without repeatedly starting from scratch.
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The value of notes appears when a puzzle stops yielding obvious placements. A list with one digit is a naked single. A digit that has only one possible position in a row, column or box is a hidden single, even if that cell’s list still contains several digits. Accurate lists also let you recognize subsets and locked candidates. More advanced methods, including X-Wing, Swordfish and XY-Wing, depend on the same lists being complete and current.
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Scan the grid before writing anything
Start with the information already printed in the puzzle. Look across each row, down each column and through each 3×3 box for missing digits and straightforward placements. If a unit is missing one digit, place it. If an empty cell has only one legal value after a quick check, place that value.
Do not fill every cell with marks automatically before this scan. An easy placement changes several rows, columns and boxes; finding it first prevents unnecessary writing and gives you a cleaner starting point. Begin candidate notation when direct scanning and ordinary singles stop producing progress, or complete the notes for unresolved cells when you need a systematic view of the remainder.
A quick scan routine
- Choose a row with several givens and list the digits it lacks.
- For each missing digit, check which empty cells in that row are allowed by their columns and boxes.
- Repeat the same check for columns and then boxes.
- Place any value that has one legal cell, and restart the scan after every placement.
This first pass is not a different solving method. It is the check that keeps your later notes small, relevant and easier to audit.
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For each unresolved cell, begin with the complete digit set 1 through 9. Cross out every digit already present in the cell’s row, column or box. The digits left over are the candidates. Write only those survivors.
Example of the three-constraint check
Suppose an empty cell lies in a row containing 1, 3, 5 and 8, a column containing 2 and 6, and a box containing 4. The union of forbidden digits is 1, 2, 3, 4, 5, 6 and 8. The legal list is therefore 7 and 9, written as 79 in the cell. If a later placement puts 9 in that row, the list becomes 7, making 7 a naked single.
The order of checking does not matter, but checking all three units does. A digit absent from the row may still be forbidden by the column or box. A note that ignores even one constraint is not a candidate; it is an error that can create false deductions later.
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Use a consistent layout
On paper, place each candidate in a predictable mini-position—often the cell’s 3×3 subgrid position corresponding to the digit. Consistent placement makes a missing or duplicated mark easier to spot. If you use compact text notation while practicing, keep the digits sorted, such as 247, rather than changing their order from cell to cell.
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Candidate notation is useful only while it reflects the current grid. Whenever you place a digit, remove that digit from every empty peer cell in its row, column and box. Whenever you eliminate a candidate because of a pattern, update the affected cells immediately.
- Write the confirmed digit in its cell.
- Inspect the other eight cells in the same row and erase that digit from their notes.
- Inspect the other eight cells in the same column and erase it there.
- Inspect the remaining cells in the same 3×3 box and erase it there.
- Check the changed units for a new naked single, hidden single, subset or locked-candidate opportunity.
Do not postpone this cleanup until the end of a chain of moves. A stale mark can make a pair appear to exist when it does not, or conceal a contradiction until much later. After every elimination, recheck the unit that changed; one removal can expose a single immediately.
Find singles before using more complex patterns
Naked singles
A naked single is a cell whose candidate list has exactly one digit. The conclusion comes from that cell alone: all other digits have been ruled out by its row, column or box. Place the remaining digit, then propagate the update through its peers.
Hidden singles
A hidden single is found by looking at a unit rather than at one cell. If a row, column or box still needs a particular digit and that digit appears in the candidate list of only one cell in the unit, that cell must contain the digit. The cell may display other marks; those alternatives are eliminated because the unit has only one position for the hidden digit.
To search for hidden singles, choose one unit and examine one digit at a time. For a row missing 2, 4 and 9, count the cells that still list 2, then the cells that list 4, then the cells that list 9. A count of one identifies the placement. Perform the same check in columns and boxes.
Use locked candidates when a digit is confined
A locked-candidate deduction occurs when all possible positions for a digit in one unit lie on a single intersecting line or inside a single intersecting box.
Pointing from a box to a line
If every candidate for digit 6 in a 3×3 box lies in the same row, 6 cannot appear in the rest of that row outside the box. The box has already reserved that row position for 6. Remove 6 from the other cells in the row, then inspect the row and the box for singles or subsets.
Claiming from a line to a box
If all candidate positions for digit 3 in a row lie inside one 3×3 box, 3 can be removed from other cells in that box outside the row. The same logic applies to a column. In both directions, the elimination is justified by the intersection: the digit’s legal positions are confined to one line-box relationship.
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Recognize naked and hidden subsets
Naked pair
A naked pair consists of two cells in the same unit whose combined candidate lists contain exactly the same two digits—and no other digits. If two cells in a row are 27 and 27, those cells must take 2 and 7 in some order. Remove 2 and 7 from every other candidate list in that row. The same rule works in a column or box.
The phrase “exactly the same two digits” matters. A cell with 27 and another with 279 do not form a naked pair, because the second cell has a third possibility. A pair also has to be confined to one unit; two matching lists in different rows provide no such deduction.
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Naked triples
A naked triple extends the same idea to three cells. Across those three cells, no more than three distinct candidates may appear, and those candidates are reserved for the trio. Remove the three digits from other cells in the unit. The individual lists need not be identical: combinations such as 12, 23 and 13 can form a triple because their union is only 1, 2 and 3.
Hidden pairs and triples
Hidden subsets are found from the opposite direction. Instead of starting with cells that have short lists, choose two or three digits and inspect where each can go in the unit. If two digits occur only in the same two cells, those cells are reserved for the pair; erase any other candidates from them. A hidden triple works the same way with three digits and three cells. This is why complete candidate lists matter: a hidden subset can be invisible if you recorded only the most obvious marks.
Escalate to advanced patterns only after simpler checks
When singles, locked candidates and subsets no longer simplify the grid, accurate notes support advanced patterns. LiveSudoku’s technique overview lists pairs, triples, X-Wing, Swordfish and XY-Wing as candidate-dependent methods. These are escalation tools, not a mandated sequence; no single technique order guarantees a solve for every difficult puzzle.
X-Wing in practical terms
An X-Wing concerns one digit appearing in exactly two candidate positions in each of two rows, with the two positions aligned in the same pair of columns. Those four positions form the corners of a rectangle. Because the digit must occupy one corner in each row, the two columns cannot contain that digit elsewhere in other rows; those outside candidates can be removed. The same pattern can be viewed with rows and columns exchanged.
Do not search for an X-Wing while an unexamined single or locked candidate is available. After an X-Wing elimination, update every affected list and rescan for simpler consequences. The elimination is only as sound as the original candidate lists; one stale mark invalidates the rectangle.
Other fish and wing patterns
Swordfish generalizes the fish idea across three rows or three columns, while XY-Wing uses a specific relationship among three two-candidate cells. Their names are less important than the discipline behind them: identify the exact candidate positions, verify that the required units intersect as the pattern demands, remove only candidates justified by that structure, and then return to ordinary scans. If you cannot explain why a candidate is eliminated, do not erase it.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Full candidate notation or selective notes?
Both notation styles are practical. The right choice depends on whether you need a complete map of the grid or a low-clutter working surface.
| Approach | How it works | Advantages | Trade-offs |
|---|---|---|---|
| Full candidate notation | Record every legal candidate in each unresolved cell. | Provides a complete comparison set for hidden subsets, naked subsets, locked candidates and fish patterns; makes omissions easier to detect. | Takes longer to write and requires more updating after every placement or elimination. |
| Selective or local notation | Mark only the candidates that seem immediately useful or the cells in the current area of attention. | Faster to enter and less visually crowded; useful during an easy phase of a puzzle. | Can hide a pattern that depends on seeing candidates across several cells or units; requires returning to fill missing lists when progress stalls. |
For a difficult puzzle, full marks are generally the safer working method because pairs, triples and fish patterns depend on seeing candidate relationships across the grid. Selective marks can be efficient when the puzzle is nearly solved or when you are deliberately concentrating on one unit. Treat this as a workflow choice, not a rule that one style always wins.
Paper notes versus digital candidate mode
| Medium | What to expect | Best practice |
|---|---|---|
| Paper | Candidate marks must be small enough to leave the final digits legible and must be erased or overwritten as the grid changes. | Use a sharp pencil and an eraser, keep digit positions consistent, and erase eliminated marks immediately rather than circling them ambiguously. |
| Digital puzzle | Candidate mode can display several notes in one cell and usually lets you toggle a digit on or off. | Confirm that you are in candidate mode before tapping; after placing a digit, verify that peer candidates were removed and that the app did not merely hide them. |
A digital candidate mode is a notation facility, not a solving guarantee. The same legality checks and audit habits apply whether marks are graphite or pixels. The available evidence establishes that such a mode exists, but it does not establish a recommendation for a particular app.
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A repeatable workflow for a hard classic 9×9
- Scan for direct placements. Check rows, columns and boxes for missing digits and obvious singles.
- Choose a notation style. Use full marks when you expect to compare many cells or look for advanced patterns; use selective marks only when the grid is simple enough to audit.
- Write legal candidates. For every unresolved cell, remove digits present in its row, column and box. Never write a digit merely because it “seems likely.”
- Place naked singles. A one-digit list is decisive. Enter it and update all peers.
- Search for hidden singles. In each row, column and box, count the positions available to each missing digit.
- Apply locked candidates. If a digit’s positions in a box are restricted to one row or column, remove that digit from the rest of the line; if a line restricts the digit to one box, remove it from the rest of the box.
- Compare subsets. Check for naked pairs and triples, then hidden pairs and triples. Remove only the candidates reserved by the verified subset.
- Escalate cautiously. Examine X-Wing or another advanced pattern only after simpler methods have been checked and no longer progress the grid.
- Propagate and rescan. After every placement or elimination, update affected notes and return to singles before looking for another pattern.
- Audit when stalled. Recalculate questionable lists against all three constraints, confirm every placed digit is legal, and look for a stale note before considering any guess.
How to audit a contradiction or stalled grid
A contradiction is evidence that something in the working state needs checking; it is not permission to turn an unexplained guess into a deduction. Use this audit sequence:
- Check every placed digit against its row, column and box.
- For a cell with no candidates, recompute its list from 1 through 9 rather than trying to “force” a digit.
- Inspect the last placement and remove it temporarily if it created an illegal duplicate.
- Recheck every candidate elimination made by a pair, triple, locked candidate or advanced pattern.
- Look for a digit with no legal position in a row, column or box; this often identifies a mistaken note or placement.
- After correcting the error, update the affected peers and resume the ordinary scan.
If the notes are consistent and no deduction follows, the puzzle may require a method beyond the patterns you have established or a carefully managed assumption. Candidate notes improve the visibility and explainability of such work, but they do not make every difficult puzzle solvable without guessing.
What candidate notes can—and cannot—do
Notes make the current possibilities explicit. They let you prove a placement from one cell, prove a hidden single from one unit, reserve digits with subsets, and justify eliminations that span intersecting units. They also make mistakes easier to locate because every mark has a stated reason for remaining.
They cannot compensate for an unchecked grid. A complete page of incorrect or stale marks is less useful than a short list that has been verified. The essential habit is therefore cyclical: calculate, compare, place or eliminate, update, and rescan. When a deduction cannot be explained from the current notes and Sudoku’s row-column-box constraints, leave the candidate in place and investigate rather than guessing.
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