Simple Coloring

Simple Coloring is a one-digit Sudoku technique that turns a connected network of conjugate pairs into two alternating logical states.

Those states are traditionally displayed as two colors.

For one candidate digit:

• every conjugate pair contains exactly one true endpoint;
• therefore its two endpoints receive opposite colors;
• connected conjugate pairs propagate that alternation through the grid.

You do not need to know initially which color is true.

Instead, you look for a relationship that is impossible under one polarity or impossible for an outside candidate under both possible polarities.

The two standard Simple Coloring eliminations are:

• Color Trap;
• Color Wrap.

Simple Coloring is especially useful when one digit forms several connected Strong Links but no short named pattern is immediately obvious.

Important accessibility rule

“Coloring” describes the traditional visual method, not a requirement to rely on color perception.

A robust digital implementation should encode the two states redundantly, for example with:

• two colors;
• A/B labels;
• distinct symbols;
• line or outline styles;
• accessible contrast.

These Guides refer to Color A and Color B so the reasoning remains understandable without depending on hue alone.

The logical structure is binary; the visual styling is only an aid.

Prerequisite: conjugate pairs

Simple Coloring propagates through Strong Links for one digit.

Suppose digit 7 appears in exactly two cells of row 2.

Those candidate 7s form a conjugate pair:

exactly one of them is true.

Assign:

• one endpoint Color A;
• the other Color B.

If one of those cells also participates in another conjugate pair in a column or box, its partner receives the opposite color again.

That is how the network grows.

A weak-only relationship does not propagate the A/B alternation.

How to build a Simple Coloring network

Choose one candidate digit, for example 7.

1. Find a conjugate pair

Look for a row, column, or box with exactly two candidate 7s.

2. Assign opposite states

Mark one candidate A and the other B.

3. Follow connected conjugate pairs

Whenever a colored candidate has another conjugate partner for 7 in a different house, color that partner with the opposite state.

4. Continue through the connected component

Repeat until no new Strong-Link partner can be reached.

5. Stop at the component boundary

Do not jump through a candidate merely because it sees a colored node. Visibility alone is Weak and does not tell you which color the new candidate should receive.

What the two colors mean

Because every coloring edge is a conjugate pair, exactly one endpoint of each edge is true.

Across one connected component there are therefore two consistent possibilities:

Case 1
Color A = true
Color B = false

or

Case 2
Color A = false
Color B = true

You do not choose between the cases at random.

You inspect the grid for a contradiction or elimination that settles part of the puzzle regardless of which case is correct.

This is why Coloring is logic, not guessing.

Color Trap

A Color Trap occurs when an uncolored candidate for the coloring digit sees:

• at least one Color A candidate;
• and at least one Color B candidate;
• within the same connected coloring component.

One of the two color polarities must be true.

So the outside candidate will see a true copy of that digit in either possible coloring state.

Therefore the outside candidate is impossible.

The key proof is:

If A is true → target conflicts with A.
If B is true → target conflicts with B.
Therefore target is false in both cases.

Color Wrap

A Color Wrap occurs when two candidates with the same color see each other.

Suppose two Color A candidates for digit 5 share a column.

If Color A were the true polarity, both cells would contain 5 in the same column — impossible.

Therefore:

Color A cannot be true.

Every Color A candidate in that connected component is false.

The opposite polarity is the surviving state of the coloring network.

Color Trap vs Color Wrap

These two deductions solve different logical problems.

Color Trap

An outside uncolored candidate sees opposite colors.

Result:

eliminate the outside candidate.

Color Wrap

Two same-color candidates inside the network conflict with each other.

Result:

eliminate that entire color polarity from the component.

A useful memory rule is:

Trap catches an outsider. Wrap breaks a color.

The mnemonic is optional; the proof should always come from visibility and truth states.

Why same-color candidates normally do not see each other

If the coloring network has been built correctly, two nodes with the same color represent the same truth polarity.

So if they also see each other for the same digit, the “that color is true” case creates an immediate Sudoku violation.

That contradiction is precisely what makes Color Wrap powerful.

Do not try to repair the coloring by swapping one isolated node. The conflict proves something about the entire connected polarity, not about one individual label.

Multiple disconnected coloring networks

A candidate digit may produce more than one disconnected conjugate-pair component.

For example:

• component 1 has A1/B1;
• component 2 has A2/B2.

Calling both first colors “A” does not create a logical relationship between A1 and A2.

Each connected component can independently flip its polarity.

So:

Color A in one disconnected component is not automatically equivalent to Color A in another.

Do not apply Color Trap or Color Wrap across disconnected networks unless another valid inference explicitly relates them.

This limitation is one reason Multi Coloring exists as a more advanced technique.

A further coloring contradiction: one polarity empties a house

Color Trap and Color Wrap are the two classical Simple Coloring outcomes you should learn first.

There is also a broader contradiction worth understanding.

Suppose one coloring polarity — for example Color A — can see every remaining position for the coloring digit inside a particular row, column, or box.

If Color A were true, all of those remaining positions would be eliminated.

The house would then have no legal place for that digit, which is impossible.

Therefore Color A must be false throughout the connected component.

Conceptually:

Assume Color A is true
→ every candidate X in one required house is eliminated
→ the house has no place for X
→ contradiction
→ Color A is false

This is a more advanced use of the same two-color logic rather than a different coloring system. It is less important for a first manual workflow than Trap and Wrap, so treat it as an extension after those two deductions are comfortable.

Simple Coloring vs Multi Coloring

Simple Coloring works inside one connected two-color component.

Multi Coloring reasons across multiple disconnected conjugate-pair components by introducing additional interactions between their color pairs.

That belongs to the Expansion layer.

For now, keep the rule simple:

one connected component, two alternating states, then Trap or Wrap.

Simple Coloring vs X-Chain

Both are one-digit techniques built on Strong/Weak inference logic.

Simple Coloring

• builds a connected network of conjugate pairs;
• assigns two alternating states;
• searches for Wrap or Trap relationships involving the network.

X-Chain

• follows one deliberate alternating Strong/Weak path;
• may cross Weak connections that are not themselves conjugate pairs;
• uses endpoint logic to produce an elimination.

A coloring network may contain several possible X-Chain paths.

Coloring is often easier when the Strong-Link graph is visually rich. X-Chains are more flexible when the useful deduction depends on a specific path through Weak connections.

Coloring is chain logic made visual

Simple Coloring is not a separate law of Sudoku.

It externalizes binary inference.

Instead of repeatedly saying:

this false → that true → this false → that true

you mark alternating candidate states and inspect the complete network.

That makes three things easier to see:

• polarity;
• contradiction;
• outside candidates that conflict with both possible polarities.

Once that feels natural, X-Chains and later AICs become much easier to understand.

How to choose a good digit for Coloring

Do not color every digit in every puzzle.

A productive candidate digit usually has:

• several conjugate pairs;
• overlaps between row, column, and box links;
• one reasonably large connected component;
• nearby uncolored candidates that may see both polarities.

A digit with only one isolated conjugate pair cannot create a meaningful coloring network by itself.

A digit with many candidates but almost no conjugate pairs may also be a poor target.

How to search for useful Simple Coloring

Use this workflow:

1. filter one candidate digit;
2. mark all obvious conjugate pairs;
3. choose a connected component with several links;
4. assign A/B states through Strong Links only;
5. check whether two same-color nodes see each other → Wrap;
6. check whether an uncolored candidate sees both colors → Trap;
7. if neither exists, leave the component and try another digit or technique.

The final two checks are where the deduction happens. Coloring the network alone is not a solving move.

A valid coloring network can be unproductive

You may correctly color ten candidates and still find:

• no same-color conflict;
• no outside candidate seeing opposite colors.

That does not mean the coloring is wrong.

It means the network currently proves no elimination.

Do not force a result. Return to the grid and search elsewhere.

This distinction between valid structure and productive deduction is important throughout advanced Sudoku.

Common mistakes

Coloring through a Weak-only relationship

A/B alternation propagates through conjugate-pair Strong Links.

A candidate merely seeing another candidate is not enough.

Comparing colors across disconnected components

Color names are local to one connected component.

Eliminating both colors after a Wrap

Only the contradictory polarity is impossible.

Eliminating only the two conflicting same-color nodes

The contradiction rejects that color throughout the connected component, not just at the visible conflict.

Treating any nearby uncolored candidate as trapped

For a Color Trap, the target must see at least one candidate of each opposite color in the same component.

Forgetting the coloring digit

Simple Coloring follows one digit only.

Do not connect different candidate values into the same simple-color network.

Relying on color alone

Use labels, symbols, or another redundant state cue for accessibility and verification.

Recognition drill

Filter one digit and ask:

Which houses contain exactly two positions for this digit?

Build one connected conjugate network.

Then ask only two questions:

Do two same-color candidates see each other?

If yes → Color Wrap.

Does an uncolored candidate see both opposite colors?

If yes → Color Trap.

If neither condition exists, stop. The network is currently unproductive.

FAQ

What is Simple Coloring in Sudoku?

Simple Coloring assigns two alternating states to a connected network of conjugate pairs for one candidate digit, then uses contradictions or outside visibility to eliminate candidates.

Is Simple Coloring one-digit only?

Yes. Standard Simple Coloring follows one candidate digit at a time.

Is Coloring guessing between two colors?

No. You do not arbitrarily choose which color is true. You derive deductions that hold because one of the two polarities must be true, or because one polarity creates a contradiction.

What is Color Trap?

An uncolored candidate sees one node of each opposite color in the same connected component, so it conflicts with whichever polarity is true and can be eliminated.

What is Color Wrap?

Two same-color candidates see each other, making that entire color polarity impossible in the connected component.

Are colors meaningful across separate networks?

No. Disconnected components can flip independently unless another valid inference links them.

Is Simple Coloring the same as X-Chain?

No. They use related logic, but Coloring manages a Strong-Link network with two states, while X-Chain follows a specific alternating Strong/Weak path.

What to learn next

Continue to X-Chain, where the same one-digit inference logic is followed as a deliberate alternating path rather than as a complete colored Strong-Link network.