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Candidate notes—also called pencil marks—turn an unsolved Sudoku cell into a precise list of digits that are still legal there. Used correctly, they reveal naked and hidden singles, pairs, triples, locked candidates, and advanced patterns such as X-Wing. Used carelessly, stale or illegal notes create deductions that only appear convincing.
The reliable method is disciplined: scan for easy placements first, write only candidates allowed by the cell’s row, column, and 3×3 box, update every affected peer after each change, and escalate to more complex patterns only when simpler checks are exhausted.
What a candidate note means
A candidate is a digit that could legally occupy an empty cell after checking all three Sudoku constraints:
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- Row: the digit is not already placed in that row.
- Column: the digit is not already placed in that column.
- 3×3 box: the digit is not already placed in that box.
A small written digit in an empty cell is therefore a working hypothesis, not a commitment to place that digit. If any of the three units already contains the digit, it is not a candidate and must not be written. Candidate notes are working memory for deductions; they do not replace checking the grid.
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LiveSudoku’s explanation of pencil marks uses this same definition and connects accurate lists to singles, locked candidates, pairs, triples, X-Wing, Swordfish, and XY-Wing. Mt. Sudoku’s hard-puzzle workflow recommends full pencil marks when basic scanning no longer produces progress.
Scan before you write
Candidate notation is most useful after the obvious information has been harvested. Start with a clean pass through the grid:
- Check every row for a missing digit that has only one possible empty cell.
- Check every column in the same way.
- Check each 3×3 box for a digit that can go in only one position.
- Place any legal single and immediately rescan the affected row, column, and box.
Do not fill the entire grid with notes while easy placements remain. Every confirmed digit changes the candidate sets of up to 20 peer cells, so early placements reduce later writing and reduce the chance of copying an outdated mark.
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On paper, reserve a consistent mini-position for each digit in a cell—for example, 1–3 across the top, 4–6 through the middle, and 7–9 across the bottom. A mark in the same position always represents the same digit. Use a pencil or erasable pen so eliminations do not leave ambiguous traces.
In a digital Sudoku, enable the app’s candidate or pencil-mark mode before entering notes. That mode is useful because several small digits can coexist in one cell; it does not make an incorrectly entered candidate legal.
How to build a complete candidate list
When scanning stops producing placements, create a list for each unresolved cell. The following procedure keeps the process reproducible.
- Start with the cell’s row. Cross out every digit already placed in that row.
- Check the column. Cross out digits already placed above or below the cell.
- Check the box. Cross out digits already present in the cell’s 3×3 box.
- Write the remainder. The digits left after all three checks are the legal candidates for that cell.
- Repeat consistently. Work through every unresolved cell, preferably box by box or row by row, so none is accidentally skipped.
For example, if a cell’s three units collectively rule out 1, 2, 4, 6, 7, and 9, its note should contain only 3, 5, and 8. The order or visual size of the marks does not matter; completeness and legality do.
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Verify as you go
After writing a cell’s list, compare each mark against the row, column, and box one more time. A duplicate given digit, a misread box boundary, or a mark copied from a neighboring cell can contaminate every later deduction. If a cell has no candidate at all, stop: either the puzzle contains an error or an earlier placement or note is wrong.
Keep notes current after every placement
Candidate work is dynamic. When you place a digit, remove that digit from every empty peer in its row, column, and box. Do not wait until the end of a page or a solving “round”; a newly reduced cell may be the next forced move.
- Circle or otherwise confirm the newly placed digit.
- Visit the rest of its row and erase that digit wherever it appears as a candidate.
- Visit the rest of its column and erase the digit there.
- Visit the other cells in its 3×3 box and erase it there.
- Recheck those three units for singles or patterns before moving elsewhere.
The same rule applies after an elimination made by a pair, triple, locked candidate, or fish pattern. An elimination can create a one-candidate cell or leave a digit with only one position in a unit. Notes that are not revised after each change are stale evidence and should not be used to justify a placement.
Find singles before advanced patterns
Naked singles
A naked single is a cell whose candidate list has been reduced to one digit. The deduction comes from the cell itself: every other digit is ruled out by its row, column, or box. Place the remaining candidate, then perform the full peer update described above.
Look for naked singles whenever you finish a candidate pass or erase a mark. A puzzle can contain several at once, and placing one may expose another in a different unit.
Hidden singles
A hidden single is different. The cell may still show several candidates, but one particular digit appears in only one cell within a row, column, or box. That digit must occupy that position because the unit has no other legal location for it.
- Choose one row, column, or box.
- For each missing digit, inspect all empty cells in that unit.
- Count the cells whose candidate lists contain the digit.
- If the count is one, place the digit in that cell and remove its other candidates.
Hidden singles are easy to miss if you read only across individual cells. They require looking vertically through a unit’s candidate lists: “Where can 5 go in this box?” rather than only “What can go in this cell?”
When singles stop: locked candidates and subsets
Locked candidates
A locked-candidate deduction uses the intersection between a 3×3 box and one of its rows or columns. If all possible positions for a digit inside a box lie on the same row, that digit cannot appear elsewhere in that row outside the box. The column version is identical.
The converse also works: if, within a row, all candidates for a digit are confined to one box, eliminate that digit from the other cells in that box outside the row. Apply the elimination only to the intersecting unit; do not remove the digit from the cells that established the restriction.
Naked pairs
A naked pair consists of two cells in one unit that together contain exactly the same two candidates, with no additional candidates in either cell. Those two digits must occupy those two cells in some order, so remove both digits from every other cell in that row, column, or box.
For instance, two cells marked {2, 7} form a pair only when each has exactly 2 and 7 and both cells belong to the same unit. A cell marked {2, 7, 9} does not qualify as one member of a naked pair. After removing 2 and 7 from the other cells, look again for singles and hidden singles.
Naked triples and hidden subsets
A naked triple extends the same logic to three cells and three digits. The three cells may show combinations such as {1, 4}, {1, 6}, and {4, 6}; together they use only 1, 4, and 6, so those digits can be removed from the unit’s remaining cells. The defining condition is the combined set of no more than three digits across exactly three cells, not that every cell list must look identical.
Hidden pairs and hidden triples are found from the opposite direction. Choose two or three digits in a unit and inspect where each can go. If those digits occur only in the same two or three cells, remove every other candidate from those cells. Hidden subsets often remain invisible when you only compare whole cell lists, which is why a systematic digit-by-digit scan matters.
Escalate to X-Wing and other advanced patterns
Accurate candidate lists can support patterns that span multiple units. X-Wing is a common example: for one digit, two rows each contain that digit as a candidate in the same two columns. Those four positions form the corners of a rectangle, allowing the digit to be removed from other cells in those two columns. The pattern can also be viewed with rows and columns exchanged.
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Do not search for X-Wing merely because the puzzle is labeled difficult. First confirm that no naked or hidden single, locked candidate, pair, or triple remains. A complicated-looking pattern is unnecessary if a simpler elimination is available, and an X-Wing built from one stale mark is invalid.
LiveSudoku lists pairs, triples, X-Wing, Swordfish, and XY-Wing as techniques that depend on dependable candidate notation. These techniques do not form a universal difficulty ranking or a guaranteed solve order. When an advanced elimination is valid, erase the affected candidates, update the intersecting units, and return to the earlier checks for newly created singles.
Full notation or selective notation?
There is no single correct amount of writing. The choice depends on whether you need a complete view of the grid or only a quick local aid.
| Approach | What you record | Strength | Trade-off |
|---|---|---|---|
| Full candidate notation | Every legal candidate in every unresolved cell | Exposes relationships across rows, columns, and boxes; useful for subsets and fish patterns | Requires more writing and more updates after each placement or elimination |
| Selective or local notation | Candidates only in cells or units currently under consideration | Faster and less visually crowded | Can hide a pattern that depends on seeing candidates in several cells |
For a genuinely hard puzzle, full marks provide the broadest evidence for deductions. Selective marks can be efficient after a clear local restriction, but before relying on them for a pattern, verify that omitted cells cannot supply another position for the digit.
Paper notes versus digital candidate mode
| Method | Practical requirement | Useful characteristic |
|---|---|---|
| Paper | Erasable marks or marks that can be cleanly overwritten | Flexible, visible notation with no software dependency |
| Digital | An input mode that allows several candidate digits in one cell | Small notes can be displayed together and edited without rewriting the grid |
The solving logic is identical in both formats. Digital candidate mode records your input; it does not necessarily validate every deduction, so continue checking the three constraints yourself.
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Use this loop whenever a difficult grid appears to stop moving:
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- Scan the changed units. After the latest placement, inspect its row, column, and box for naked and hidden singles.
- Complete missing notes. If the grid still stalls, finish legal candidate lists in the unresolved cells of those units, then expand to the rest of the grid if needed.
- Check digits within each unit. Look for hidden singles, locked candidates, and naked pairs before considering larger subsets.
- Check subsets. Search for naked triples and hidden pairs or triples by comparing both cells and digits.
- Inspect broader patterns. Only after the simpler checks fail, examine candidates across multiple rows and columns for X-Wing or another documented pattern.
- Apply one deduction at a time. Erase the candidate, update every affected peer, and restart the scan from the changed units.
- Audit instead of guessing. If no legal deduction follows, verify the notes and all placed digits before making any assumption.
This is a priority order, not a promise that every puzzle can be completed without guessing. Candidate notes make deductions explainable and help expose contradictions; they do not guarantee that a particular technique sequence will solve every classic 9×9 puzzle.
How to audit notes when progress stops
A contradiction is diagnostic information. If a cell has no candidates, or a unit has no possible position for a missing digit, work backward before attempting a guess.
- Recheck the original givens and every placed digit in the affected row, column, and box.
- Recalculate the cell’s candidates from scratch instead of trusting an old list.
- Look for a candidate that was erased from a peer but should have remained.
- Confirm that a claimed pair, triple, or locked candidate met its exact conditions.
- Undo the most recent uncertain placement or elimination and rescan.
A stale note can create a false single; a mistaken elimination can create an impossible unit later. Correcting the record is more reliable than turning an unexplained contradiction into a guess.
Common candidate-note mistakes
Writing every digit from 1 to 9
Those marks are not candidates unless all three units permit them. Starting with an unrestricted list defeats the purpose of pencil marks and makes hidden patterns harder to see.
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Leaving eliminated marks in place
A note that was legal ten moves ago may be illegal now. Update peers after each placement and after every pattern-based elimination.
Confusing a pair with a near-pair
Two cells must contain exactly the same two candidates for a naked pair. Extra marks invalidate the deduction; remove them only after proving they are impossible by another rule.
Reading only cell lists
Hidden singles and hidden subsets are found by tracking where a digit can appear in a whole unit. Scan by digit as well as by cell.
Jumping to a complex pattern
Advanced techniques consume attention and depend on cleaner data. Recheck singles, locked candidates, and subsets first, then use the simplest valid elimination available.
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- Have you scanned rows, columns, and boxes for straightforward singles?
- Does every written candidate pass the row, column, and box test?
- After each placement, did you remove that digit from all 20 peers?
- Did you inspect both naked singles and hidden singles?
- Did you test locked candidates before searching for fish patterns?
- Do any proposed pairs or triples meet their exact cell-and-digit conditions?
- After an elimination, did you rescan the affected units?
- If the grid contradicts itself, did you audit notes and placements before guessing?
Consistent notes convert a difficult Sudoku from a visual tangle into a record of constrained possibilities. The strongest habit is not writing more marks; it is keeping every mark legal, current, and tied to a deduction you can explain.
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