Sudoku Logic

Sudoku logic is the process of turning a small set of constraints into conclusions that must be true.

At the beginning of a puzzle, the clues tell you very little cell by cell. The power comes from combining overlapping restrictions:

• every row needs digits 1–9 exactly once;
• every column needs digits 1–9 exactly once;
• every 3×3 box needs digits 1–9 exactly once.

A logical solving step extracts new information from those constraints.

Sometimes that information is a digit you can place.

Sometimes it is only:

This candidate is impossible.

That is still real progress.

Four kinds of information

Most deductions in the current learning path can be understood as one of four information types.

1. Forced value

This cell must be X.

Examples:

• last missing digit;
• Naked Single;
• Hidden Single.

2. Forbidden value

This cell cannot be X.

Examples:

• Locked Candidates;
• subsets;
• Fish;
• Wings;
• chain eliminations.

3. Reservation

These N digits belong to these N cells, although their exact order is unresolved.

Examples:

• Naked Pair;
• Hidden Pair;
• Triples.

4. Implication

If this candidate has one state, another candidate is forced into another state.

Examples:

• Strong/Weak Links;
• Coloring;
• X-Chains;
• XY-Chains.

Candidates are a model of uncertainty

A candidate is not a weak guess.

It means:

This digit has not yet been ruled out by the current state.

For a cell r4c7, the candidate set might be:

{2,5,8}

As deductions accumulate, that domain shrinks.

If it becomes:

{8}

you have a Naked Single.

From a mathematical perspective, candidate management is a human-readable form of constraint propagation: every solved value restricts its peers, and every derived restriction can trigger further restrictions.

Cell-first reasoning

Cell-first logic asks:

What can this cell contain?

Typical techniques:

• Naked Single;
• Naked Pair;
• Naked Triple;
• XY-Wing / bivalue relationships.

Example:

Two cells in one row are both {3,9}.

Those cells reserve 3 and 9.

Other cells in the row cannot use either digit.

The exact order is unknown.

The reservation is still certain.

Digit-first reasoning

Digit-first logic asks:

Where can this digit go?

Typical techniques:

• Hidden Single;
• Locked Candidates;
• Hidden subsets;
• X-Wing;
• Swordfish;
• Simple Coloring;
• X-Chain.

Example:

Every candidate 6 in one box lies on row 4.

The exact 6 cell is unknown.

But the box guarantees that row 4 receives its 6 inside the box.

So candidate 6 can be removed from row 4 outside the box.

This is a useful theme throughout Sudoku:

Exact placement can remain unknown while a weaker positional fact is already strong enough to eliminate candidates.

Local logic and distributed logic

Local deductions

Their proof lives mainly in one cell, unit, or intersection:

• Singles;
• Locked Candidates;
• Pairs;
• Triples.

Distributed deductions

Their proof links information across more distant units or cells:

• X-Wing;
• Swordfish;
• Skyscraper;
• W-Wing;
• Coloring;
• Chains.

Advanced Sudoku does not introduce new row/column/box rules.

It combines the same constraints across a larger dependency structure.

Counting is one of Sudoku's deepest recurring ideas

Several apparently unrelated techniques are set-counting arguments.

Subset counting

N cells ↔ N digits

If three cells can contain only three digits between them, those digits are reserved.

Fish counting

N base units → N cover units

If N base units need N copies of one digit and all source positions are covered by N opposite units, those cover units are consumed by the bases.

Hidden subset counting

N digits → N possible cells

If N required digits have no positions outside N cells, those cells are reserved for them.

The objects change.

The logic of constrained sets stays surprisingly similar.

Implication logic

Advanced solving becomes easier when candidate relationships are written as implications.

Strong Link

Two candidate states cannot both be false.

A false
→ B true

Weak Link

Two candidate states cannot both be true.

A true
→ B false

Alternating those roles allows a state to propagate:

false
→ true
→ false
→ true

A useful Chain turns that propagation into a conclusion about an endpoint, target candidate, or contradiction.

Contradiction is a proof method

Suppose you temporarily assume:

candidate A is true.

Then every following inference is forced.

If the chain leads to an impossible state, the assumption was false.

This is not the same as randomly entering A and hoping.

The temporary assumption is used to prove the opposite state.

Possible contradictions include:

• duplicate solved digit in a unit;
• an unsolved cell with no candidates;
• a required digit with no legal position in a unit;
• an impossible chain state.

Why “legal” is weaker than “solved”

A digit may be legal in a cell because it creates no immediate conflict.

That does not mean the digit is logically determined there.

Example:

r5c4 {3,8}

Both candidates may be legal.

Choosing 3 merely because it fits is not a deduction.

A logical placement must explain why 8 has been eliminated or why 3 is otherwise forced.

This distinction is central to both the Rules and the Learn system.

Useful deduction vs valid pattern

A pattern can exist without changing the board.

For example:

• a Naked Pair whose digits appear nowhere else in the unit;
• an X-Wing whose cover columns contain no outside candidate;
• a Chain whose endpoints have no common-peer target.

The structure can be valid.

The solving move is unproductive.

VeyraPlay Practice should eventually test this distinction because it prevents pattern collecting from replacing actual solving.

The dependency cascade

Sudoku rarely progresses as isolated named techniques.

A realistic sequence might be:

X-Wing
↓
candidate elimination
↓
Hidden Single
↓
placement
↓
peer update
↓
Locked Candidates
↓
Naked Single

The advanced move matters because of the simpler information it exposes.

That is why VeyraPlay's strategy Guides repeatedly say:

After any elimination, re-scan for simple logic.

Correctness solver vs human-style solver

Software can answer different questions.

Correctness / solution-count solver

Does a solution exist?

Is there exactly one?

It may use:

• backtracking;
• exact cover;
• SAT/constraint search;
• another complete algorithm.

Human-style analyzer

What understandable deduction is available next?

It attempts the technique library in a defined order and records:

• placements;
• eliminations;
• required techniques;
• dependencies;
• difficulty features.

VeyraPlay benefits from both.

Using one solver for every job would blur correctness, pedagogy, and difficulty.

Uniqueness is an additional premise

Unique Rectangle does not follow only from the three local Sudoku constraints.

It additionally assumes:

This puzzle has exactly one solution.

That premise can be valid and useful, but it must be explicit.

VeyraPlay should only surface uniqueness deductions when the puzzle has been verified unique and the solving mode allows them.

What “logical” means in VeyraPlay

For this product, a logical deduction should be:

• derived from known puzzle constraints and any explicit extra premise;
• reproducible from the current state;
• explainable step by step;
• independently verifiable by the engine;
• productive if it is presented as the next solving move.

The technique name is secondary.

The proof is the important part.

FAQ

Is candidate elimination a real Sudoku move?

Yes. Intermediate and Advanced solving often progresses through eliminations before another digit can be placed.

Are Advanced techniques based on different rules?

Normally no. They combine the same constraints across more cells or units. Uniqueness techniques explicitly add the verified one-solution premise.

Is contradiction guessing?

Not when a temporary assumption is used inside a valid proof and the conclusion is forced.

Why can two solvers find different next moves?

Several deductions can be available at the same time. A human-style solver needs a priority model, usually preferring simpler productive logic.

Does every valid pattern need to be used?

No. If it produces no new information or a simpler deduction is available, it may be better to ignore it.

What to learn next

For the practical question “Do I ever need to guess?”, continue to Do You Have to Guess in Sudoku?

For the mathematical representation of the puzzle, continue to Sudoku and Mathematics.