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Simple Coloring

Learn how Simple Coloring uses conjugate pairs and two colors to find Color Trap and Color Wrap eliminations.

Simple Coloring follows one candidate digit through a network of conjugate pairs and assigns two alternating states — traditionally shown as two colors.

Every Strong Link flips the state.

If one color represents true, the other represents false.

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

Instead, you look for consequences that are valid under either possible color assignment.

The two main outcomes taught here are:

  • Color Trap;
  • Color Wrap.

Important accessibility rule

Coloring is a logical name, not a reason to make the UI depend only on color perception.

VeyraPlay should combine:

  • two colors;
  • symbols/patterns;
  • link lines;
  • accessible labels such as A/B.

Editorially, we can say Color A / Color B while designing a redundant visual representation.

Build the coloring network

Choose one digit, for example 7.

Find a conjugate pair:

  • exactly two candidate 7s in one unit.

Assign:

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

Now continue through connected conjugate pairs.

Whenever a colored candidate has a Strong-Link partner:

  • partner gets the opposite color.

Continue until that connected component cannot grow.

What the colors mean

Because each colored edge is Strong:

  • if A is false → B is true;
  • if B is false → A is true.

Across a consistent connected coloring component, one polarity corresponds to true and the other to false.

Conceptually:

Case 1
A = true
B = false

or

Case 2
A = false
B = true

The useful deductions come from relationships that make one of those cases impossible or make an outside candidate impossible in both.

Color Trap

A Color Trap occurs when an uncolored candidate sees:

  • one Color A node;
  • one Color B node.

One color polarity must be true.

So one of those two colored candidates must contain the digit.

The uncolored candidate sees whichever one is true.

Therefore the uncolored candidate cannot contain the digit.

Color Wrap

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

If Color A were the true polarity, both same-color candidates would be true.

But they share a unit and cannot both contain the digit.

Therefore Color A cannot be the true polarity.

Every Color A candidate in that connected component is false.

Why Color Trap and Color Wrap are different

Trap

An outside uncolored candidate is eliminated because it sees opposite colors.

Wrap

An entire color polarity is eliminated because two same-color nodes conflict.

That distinction should be visible in the final components.

Multiple disconnected color networks

Suppose candidate 6 forms two separate conjugate-pair components.

Color A in component 1 has no automatic relationship to Color A in component 2.

The labels are local.

Do not compare colors across disconnected networks unless another valid inference connects them.

This is an important practical source of false coloring deductions.

Simple Coloring vs X-Chain

Both use one digit.

Simple Coloring

Build a connected Strong-Link network and search it for contradictions/interactions.

X-Chain

Follow one specific alternating Strong/Weak path and use its endpoints.

Coloring can be easier visually when many conjugate pairs form a clean network.

X-Chains are more flexible because they deliberately include Weak Links between Strong-Link segments.

Coloring is chain logic made visual

Simple Coloring is not a separate mysterious law.

It is a way to externalize binary inference states.

This is valuable for Learn because the user can see:

  • alternating states;
  • conflict;
  • elimination.

Then Chains can reuse the same logic without relying on the color network.

How to search for useful Coloring

  1. Choose one digit.
  2. Find a digit with several conjugate pairs.
  3. Build one connected component.
  4. Check for same-color conflicts → Wrap.
  5. Check uncolored candidates seeing both colors → Trap.
  6. If neither exists, the component is currently unproductive.
  7. Try another component/digit or return to a simpler technique.

Common mistakes

Coloring through a Weak-only relationship

The A/B alternation propagates through Strong Links.

Comparing colors across disconnected components

Invalid without another connection.

Eliminating both colors after a Wrap

Only the contradictory polarity is false.

Treating any uncolored candidate near the network as trapped

It must see opposite colors.

Relying on color alone in the UI

Accessibility requires redundant state encoding.

Recognition drill

With one digit filtered:

Which units contain exactly two positions?

Build the conjugate network first.

Only then ask:

Does one color collide with itself, or does an outside candidate see both colors?

FAQ

Is Simple Coloring one-digit only?

Yes.

Is Coloring guessing between two colors?

No. You do not choose one arbitrarily; you use consequences common to the two possible states or contradictions that eliminate one polarity.

What is Color Trap?

An uncolored candidate sees opposite colors and is impossible.

What is Color Wrap?

Same-color candidates see each other, proving that polarity false.

What to learn next

Continue to X-Chain, which expresses one-digit inference as a deliberate alternating path instead of a colored Strong-Link network.

concepts · advancedStrong Links in SudokuUnderstand Strong Links, conjugate pairs and the false-implies-true inference used by W-Wings, Coloring, X-Chains and other advanced techniques.concepts · advancedWeak Links in SudokuUnderstand Weak Links in Sudoku, how they differ from Strong Links, and how true-implies-false relationships power chain solving.concepts · advancedSudoku ChainsLearn how Sudoku Chains alternate Strong and Weak Links to propagate logical states, prove endpoint conclusions and eliminate candidates.techniques · expertX-ChainLearn how X-Chains alternate Strong and Weak Links for one digit and eliminate candidates that see both chain endpoints.