01 · Foundation
What logical solving means
A standard Sudoku contains nine rows, nine columns, and nine 3×3 boxes. Each unit must contain the digits 1 through 9 exactly once. The printed digits are givens. Empty cells must be resolved from the restrictions created by those givens and later placements.
Solving logically means that each placement or candidate removal can be justified from the current grid. You are not choosing between two possibilities and hoping one works. You are proving that a digit must go somewhere—or cannot go somewhere—before you act.
The Orbace rule
Before placing a number, finish this sentence: “This cell must be ___ because…” If the reason is only “it looks likely,” keep looking. A candidate is a possibility; a solved digit is a conclusion.
02 · Workflow
A repeatable solving loop
Strong solvers do not scan randomly. They move through a short cycle, returning to simpler observations whenever the grid changes.
Scan completed digits
Choose a digit that already appears several times. Trace its rows and columns through neighboring boxes to identify where another copy can or cannot fit.
Inspect constrained units
Look at rows, columns, and boxes with many filled cells. Ask which digit is missing and which empty cells remain possible.
Add candidates selectively
When direct scanning slows, write possible digits in unresolved cells. Keep notation accurate and update it after every placement.
Find a pattern
Look for singles first, then locked candidates, subsets, and advanced patterns. Make one justified change.
Rescan immediately
An elimination may create a single elsewhere. Do not keep searching for something advanced when a simple placement has become available.
Verify before continuing
Check that the placed digit does not duplicate another in its row, column, or box and that removed candidates were affected by the pattern.
03 · Beginner
Strategies that place digits directly
Full house
A row, column, or box has only one empty cell. The missing digit must occupy that cell. List the digits already present or notice the one absent from 1–9.
Row contains 1, 2, 3, 4, 5, 6, 8, 9 → missing digit is 7.Naked single
A cell has only one legal candidate after considering its row, column, and box. The cell is “naked” because its own candidate list reveals the answer.
Question to ask
“Which digits are blocked by this cell’s three units, and is only one possibility left?”
Hidden single
A digit can go in only one cell within a row, column, or box, even though that cell may contain several candidates. The single is hidden among the cell’s other possibilities.
How to spot it
Choose one digit and scan every candidate position in a unit. If the digit appears only once, place it there.
Crosshatching
Use existing copies of one digit in neighboring rows and columns to rule out cells in a 3×3 box. When eight boxes already contain a 6, for example, the row and column restrictions often reveal the ninth 6 efficiently.
Beginner milestone
Before moving on, become comfortable distinguishing a naked single—one candidate in a cell—from a hidden single—one location for a digit in a unit. That distinction supports nearly every later technique.
04 · Intermediate
Strategies that remove candidates
Intermediate methods often do not place a digit immediately. Their value is in eliminating possibilities until a simpler technique becomes available.
Pointing pair or triple
Inside one 3×3 box, all candidates for a digit lie in the same row or column. Because the digit must appear somewhere in that box, it cannot appear elsewhere along that row or column outside the box.
In the upper-left box, candidate 5 appears only in r2c1 and r2c3 → remove 5 from other cells in row 2.Claiming pair or triple
The same relationship viewed in reverse: within a row or column, all candidates for a digit fall inside one box. That digit is therefore claimed by the box-line intersection and can be removed from other cells in the box.
Naked pair
Two cells in one unit contain the same two candidates and no others—for example, {2,8} and {2,8}. Those digits must occupy the two cells in some order, so 2 and 8 can be removed from every other cell in that unit.
A naked triple and naked quad use the same idea with three or four cells whose combined candidates are restricted to the same number of digits.
Hidden pair
Two digits appear as candidates in exactly the same two cells within a unit, even if those cells currently show other candidates. Those two cells must contain the pair, so their other candidates can be removed.
Hidden triples and quads extend the same logic, but they are harder to see and less common.
Remote pair
A chain of cells containing the same naked pair alternates between the two digits. Any cell that sees both ends of an even alternating chain cannot contain either digit common to the contradiction created there. Treat this as an introduction to chain reasoning rather than a first choice.
05 · Advanced
Patterns across several units
Advanced strategies usually connect candidate positions across rows, columns, or boxes. Mark candidates carefully; an incorrect note can create a pattern that does not really exist.
X-Wing
For one digit, two rows each contain exactly two candidate positions, and those positions align in the same two columns. The digit must occupy opposite corners of the resulting rectangle. Therefore, that digit can be removed from every other cell in the two columns.
Candidate 7 occurs in row 2 at c3/c8 and row 6 at c3/c8 → remove 7 from other cells in columns 3 and 8.The pattern also works with rows and columns reversed.
Swordfish
A Swordfish extends X-Wing logic to three rows and three columns. In three rows, a candidate is confined to two or three positions that collectively occupy the same three columns. The candidate must be placed once in each selected row, so it can be removed from other cells in those columns.
Search systematically by digit. Do not require all three rows to have identical positions; the candidates only need to be contained within the same three columns.
XY-Wing
An XY-Wing uses three bivalue cells: a pivot with candidates XY and two pincers with XZ and YZ. The pivot sees both pincers. Whichever value the pivot takes, one pincer must become Z, so Z can be removed from any cell that sees both pincers.
Pivot {2,5}; pincers {2,8} and {5,8} → a shared peer of both pincers cannot be 8.Unique rectangle
In puzzles intended to have one solution, certain four-cell arrangements of the same two candidates would permit two interchangeable solutions. When three corners of a rectangle contain only the same pair and the fourth has extra candidates, those extra candidates may be necessary to avoid the deadly pattern.
Use uniqueness-based reasoning only when the puzzle is known to have exactly one solution and after learning the precise rectangle type involved.
Simple coloring
Choose one candidate digit and link positions that form strong pairs—places where the digit can occur in exactly one of two cells in a unit. Color the alternating chain with two colors. If two cells of one color see each other, that color is false. If an uncolored candidate sees both colors, it can be removed.
Alternating inference chains
An alternating inference chain connects strong links (“if this candidate is false, that one must be true”) and weak links (“these two candidates cannot both be true”). If both possible states of the first candidate lead to the same conclusion, that conclusion can be applied.
This is a framework rather than one visual shape. Learn it after X-Wing and XY-Wing, when candidate relationships already feel natural.
06 · Notation
Use candidates as evidence, not decoration
Candidate notes are useful only when they remain accurate. Writing every possibility can clarify a hard puzzle, but overcrowded notes can hide the relationships you need.
- Start late: scan for full houses and singles before filling every candidate.
- Update immediately: after placing a digit, remove it from all peers in the same row, column, and box.
- Scan by digit: for locked candidates and fish, temporarily focus on one candidate across the grid.
- Separate facts from tests: do not enter a speculative choice as if it were a proven placement.
- Use notes consistently: if you only mark selected candidates, remember that an unmarked possibility may still exist.
A calm notation habit
Pause after every placement. Update its three units, then look for a new single before continuing. This small rhythm prevents most candidate errors.
07 · Recovery
What to do when the grid stops moving
- Check the last few placements. Confirm that each had a logical reason and did not duplicate a digit.
- Rebuild uncertain candidates. Choose the densest row, column, or box and verify every note against its peers.
- Return to singles. A previous elimination may have created one that you passed over.
- Scan one digit at a time. Follow all remaining 1s, then 2s, through 9s. This makes locked candidates and fish more visible.
- Check intersections. Look where a box meets a row or column for pointing and claiming patterns.
- Search subsets. Compare candidate groups within constrained units.
- Step away. A short break is a legitimate solving technique. Fresh attention often sees a restriction that effort was obscuring.
A hint should ideally name the technique or area to inspect before revealing a digit. If you use an Orbace replay, return to the first move where your path diverged rather than studying only the final correction.
08 · Difficulty
Technique difficulty is not player ability
A puzzle’s rating should describe the hardest logical step required by its intended solve path. It does not measure intelligence, and solving time alone does not determine difficulty.
| Level | Common required techniques | Best learning goal |
|---|---|---|
| Easy | Full houses, crosshatching, naked singles, hidden singles | Accurate scanning and confident placement |
| Medium | Singles plus pointing, claiming, and basic pairs | Candidate maintenance and eliminations |
| Hard | Multiple subset types, locked candidates, longer interactions | Recognizing when eliminations create simpler moves |
| Expert | X-Wing, XY-Wing, coloring, or comparable patterns | Tracking candidate relationships across units |
| Extreme | Swordfish, chains, combined advanced techniques | Constructing and verifying multi-step logic |
09 · Practice
A three-stage learning plan
Notice
Study one technique with a completed example. Identify the exact restriction that makes the elimination valid.
Isolate
Practice puzzles known to contain that technique. Do not worry about time; name the pattern before applying it.
Integrate
Return to mixed puzzles where the technique is not announced. Replay your Su-Pu afterward and find the decisive step.
A practical weekly rhythm
- Days 1–2: two comfortable puzzles, focusing on clean scanning.
- Day 3: study one new technique and reproduce its logic in your own words.
- Days 4–5: solve two targeted practice puzzles using that technique.
- Day 6: try one puzzle slightly above your usual level.
- Day 7: replay a Su-Pu and identify one efficient move and one missed opportunity.
10 · Questions
Sudoku strategy FAQ
Should Sudoku ever require guessing?
A well-constructed standard Sudoku with an intended logical solution should not require a blind guess. Very hard puzzles may require chain reasoning that resembles testing possibilities, but a valid chain proves a conclusion without committing an unsupported digit to the grid.
When should I start writing candidates?
Start after direct scanning and simple singles stop producing progress. On easier puzzles, selective notes may be enough. On hard puzzles, complete and accurate candidates make subsets and advanced patterns easier to verify.
What is the difference between a naked and hidden pair?
A naked pair is visible in two cells that contain only the same two candidates; remove those digits from other cells in the unit. A hidden pair consists of two digits that occur only in the same two cells; remove other candidates from those two cells.
Which advanced strategy should I learn first?
Learn X-Wing after you are comfortable with locked candidates and pairs. It extends the same idea—restricted candidate positions—across two rows and two columns. XY-Wing is a good next step.
Why do I keep reaching contradictions?
The usual cause is an earlier unsupported placement or an outdated candidate. Review the first point where certainty became assumption. Rebuild candidates in the affected row, column, and box rather than continuing from a damaged grid.
Does solving faster mean I am improving?
Sometimes, but speed is only one signal. Fewer corrections, clearer explanations, better candidate accuracy, and recognizing a technique without prompting are stronger measures of durable progress.