XY-Chain

An XY-Chain is a Sudoku chain built, in its standard form, entirely from bivalue cells — cells containing exactly two candidates.

The candidate value changes as the chain travels.

Each step alternates between:

• a Strong Link inside a bivalue cell;
• a Weak Link between peer cells that share one candidate.

The first and final cells both contain the same endpoint candidate Z.

The chain proves:

At least one endpoint must be Z.

Therefore candidate Z can be eliminated from any outside cell that sees both endpoint cells.

XY-Chain generalizes the same bivalue propagation that first appears in XY-Wing.

The basic XY-Chain structure

Consider four bivalue cells:

A {3,9}
B {9,8}
C {8,2}
D {2,3}

For this to form a chain:

• A sees B;
• B sees C;
• C sees D;
• each neighboring pair shares the linking candidate shown in the sequence.

Candidate 3 is the endpoint digit because it appears in both A and D.

Now trace the implications.

If A is not 3

A is bivalue, so:

A = 9.

A sees B on candidate 9, so:

B ≠ 9.

B is bivalue, so:

B = 8.

B sees C on 8:

C ≠ 8 → C = 2.

C sees D on 2:

D ≠ 2 → D = 3.

Therefore:

A not 3 → D = 3.

Read the same chain from the opposite end:

D not 3 → A = 3.

So A and D cannot both be non-3.

At least one endpoint contains 3.

Why every standard XY-Chain cell is bivalue

The chain depends on a Strong inference inside each cell.

For:

{3,9}

if 3 is false, 9 is forced.

But if the cell were:

{3,9,5}

then not-3 would leave:

• 9;
• or 5.

No individual partner is forced.

That breaks the standard XY-Chain propagation.

More general AICs can travel through richer structures, but those are not standard XY-Chains.

The two link types inside an XY-Chain

XY-Chain becomes much easier to verify when you separate the two link sources.

Strong Link inside a cell

A bivalue cell {X,Y} contains exactly one of its two candidates.

So:

not X → Y
not Y → X

Weak Link between cells

Suppose:

A {3,9}
B {9,8}

and A sees B.

Both cells cannot contain 9 because they share a house.

So:

A = 9 → B ≠ 9

That inter-cell candidate 9 relationship is Weak.

Then B's bivalue Strong Link turns B ≠ 9 into B = 8.

This repeated Weak-between / Strong-within structure lets the candidate value walk through the grid.

The endpoint elimination

Suppose the chain starts in cell A {Z,X} and ends in cell F {Y,Z}.

The proof establishes:

A = Z or F = Z.

Now take an outside target candidate Z.

If the target sees A, it cannot be Z when A = Z.

If the target also sees F, it cannot be Z when F = Z.

Those two cases cover every valid endpoint state.

Therefore:

the target candidate Z is false.

The target does not need to see the internal chain cells.

It must see both endpoint cells for the shared endpoint digit.

Step-by-step board example

Before reading the board, verify:

1. every chain cell is bivalue;
2. every consecutive pair of cells sees each other;
3. consecutive cells share the intended linking candidate;
4. the endpoint cells share candidate Z;
5. the outside target Z sees both endpoints.

XY-Wing is the shortest familiar introduction

XY-Wing uses three bivalue cells:

Pivot {X,Y}
Pincer 1 {X,Z}
Pincer 2 {Y,Z}

The logic can be written as a short XY-Chain from one Z endpoint to the other through the pivot relationships.

That gives a natural learning progression:

XY-Wing
→ understand bivalue propagation
→ extend the path through more bivalue cells
→ XY-Chain

If a three-cell pattern is clearly an XY-Wing, use that familiar name. XY-Chain becomes useful when the path contains additional bivalue cells or less obvious geometry.

XY-Chain vs X-Chain

Both techniques use alternating inference and can end with the same style of common-peer elimination.

X-Chain

• one candidate digit remains fixed;
• Strong Links usually come from conjugate pairs in houses;
• Weak Links connect same-digit peer candidates.

XY-Chain

• candidate digits change along the path;
• Strong Links are inside bivalue cells;
• Weak Links connect peer cells through a shared candidate.

A useful mental shortcut is:

X-Chain follows a digit. XY-Chain follows bivalue cells.

XY-Chain vs Remote Pair

A Remote Pair uses a chain of bivalue cells that all contain the same two candidates, for example {2,8} repeated through the chain.

XY-Chain allows the candidate pair to change from cell to cell:

{3,9} → {9,8} → {8,2} → {2,3}

So Remote Pair can be viewed as a more specialized repeating-pair structure, while XY-Chain is the broader changing-candidate bivalue chain.

Remote Pair is planned for the Expansion layer.

How to find XY-Chains

XY-Chains are easiest to search cell-first.

1. Highlight bivalue cells

A dense bivalue region is a better candidate graph than a puzzle with only a few isolated bivalue cells.

2. Choose a start cell

Suppose the start cell is {3,9}.

Pick 3 as the candidate you hope to recover at the far endpoint.

3. Follow the other candidate

Search for a peer bivalue cell containing 9.

4. Switch to that cell's other candidate

If the next cell is {9,8}, the chain now carries 8 forward.

5. Find a peer bivalue cell containing 8

Continue the same process.

6. Stop when the endpoint candidate returns

You are looking for a final bivalue cell containing 3.

7. Check common peers of the two endpoint cells

If an outside candidate 3 sees both endpoints, the chain may be productive.

8. Verify the whole chain in both directions

Do this before eliminating anything.

Geometry matters only through visibility

XY-Chains do not require a fixed visual shape.

Consecutive cells may connect through:

• rows;
• columns;
• boxes.

What matters is that every shared linking candidate creates a valid Weak Link because the two cells see each other.

Two bivalue cells with a shared candidate but no peer relationship are not connected in an XY-Chain.

Candidate sequence alone is never enough.

Avoid looping aimlessly

A network of bivalue cells can branch quickly.

Without discipline, manual XY-Chain searching becomes a depth-first graph search with little human value.

Prefer:

• short paths;
• endpoints near many potential targets;
• candidates that recur after a few cells;
• clearly visible peer relationships.

Stop when:

• the endpoint digit returns and gives a target;
• the path becomes difficult to verify;
• you revisit the same candidate state;
• or a simpler technique becomes available.

The goal is a deduction, not exhaustive graph traversal.

Can an XY-Chain repeat cells or candidate states?

A useful manual chain should not loop back through the same logical state simply to become longer.

Revisiting a state usually means:

• you have created a cycle;
• the path can be shortened;
• or you are moving toward a more general loop/AIC interpretation.

For the standard open XY-Chain taught here, choose a simple path from one endpoint to another.

Strong Links between chain cells do not invalidate the chain

Sometimes two consecutive bivalue cells share a candidate in a house where that candidate appears only twice.

That inter-cell relationship is therefore Strong and Weak.

The XY-Chain needs the Weak role there:

one true → the other false.

A relationship that is Strong as well can still supply that Weak role because the candidates cannot both be true.

This is the same flexibility used in other alternating chains.

A valid XY-Chain can be unproductive

You may correctly connect several bivalue cells and return to endpoint candidate Z, but find no outside Z candidate that sees both endpoints.

Then the standard common-peer elimination has no target.

The chain is logically valid but currently unproductive.

Do not eliminate from a cell seeing only one endpoint.

Common mistakes

Using a trivalue cell

A standard XY-Chain requires the pairwise Strong implication inside every chain cell.

Connecting cells that do not see each other

Shared candidate values are not enough. Consecutive cells must be peers for that linking candidate.

Forgetting which candidate carries the chain forward

Inside each bivalue cell, you enter through one candidate and leave through the other.

Ending on a different candidate

The standard endpoint elimination requires the first and final cells to share the same candidate Z.

Eliminating from a target that sees only one endpoint

The other endpoint case leaves that target possible.

Searching every bivalue path in the puzzle

Prefer short, purposeful chains with plausible endpoint targets.

Treating XY-Chain as a fixed shape

It is an inference path, not a geometric template.

Recognition drill

Start with:

{3,9}

Choose 3 as endpoint candidate.

Then say the path aloud:

not 3 → 9; 9 eliminates peer 9 → other value; that value eliminates peer value → other value...

Continue until you reach another cell containing 3.

Then ask:

Which candidate 3s see both endpoint cells?

If you cannot state every shared-candidate peer relationship, the chain is not yet proven.

FAQ

What is an XY-Chain in Sudoku?

An XY-Chain is a chain of bivalue cells in which Strong Links operate inside cells and Weak Links connect peer cells through shared candidates.

Do all XY-Chain cells need exactly two candidates?

Yes for the standard XY-Chain taught here. More general AIC structures can use richer nodes, but they are different techniques.

Can an XY-Chain use many different digits?

Yes. The candidate value can change at every cell as long as each consecutive pair forms the required shared-candidate Weak connection.

Is XY-Wing an XY-Chain?

It can be represented as the short three-cell introductory form of XY-Chain logic.

What candidate gets eliminated?

The candidate shared by the two endpoint cells, from any outside cell that sees both endpoints.

Do the endpoint cells need to see each other?

No. The elimination depends on the target seeing both endpoints. The endpoints themselves may be far apart.

What is the difference between XY-Chain and Remote Pair?

Remote Pair repeats the same two candidates through its cells. XY-Chain allows the candidate pairs to change along the path.

What to learn next

After X-Chain and XY-Chain, the natural generalization is Alternating Inference Chains (AICs), which will be added during Expansion and can mix candidate values and link structures more freely.