Advanced technique · Chains · 04

XY-Chain

Bivalue cells pass a two-candidate relationship from one end to the other.

AdvancedChainsElimination

The idea

BIVALUE CELLS PASS A COMMON CANDIDATE FROM LINK TO LINK.

An XY-Chain, also know as a Y-Chain, uses only bivalue cells, but unlike a Remote Pair, the value pairs can be different. Each cell shares one candidate with the next cell, allowing the chain to pass from one candidate pair to another.

The chain starts and ends with two bivalue cells sharing a same number. That proves the number must occur in one of the endpoints, so it can be removed from cells that see both ends.

The question: Can a sequence of bivalue cells pass from one pair to another and return to the same endpoint number?

Walkthroughs

FOLLOW THE CANDIDATE HANDOFF

At every bivalue cell, one number is carried forward and the other is excluded by the previous link. Keep the chain readable by naming the dight that changes at each step.

Walkthrough

Example 01 — A simple XY-Chain

Consider a five cells chain starting from cell r1c6 with numbers 3 and 4, traveling through cell r6c6, r3c6, r3c5 and ending at cell r9c5, if cell r1c5 is not number 3, then cell r6c6 must be number 7, cell r3c6 must be number 5, cell r3c5 must be 7, resulting cell r9c5 to be number 3; because one of chain's ending cells r1c6 or cell r9c5 must be number 3, therefore cell r1c5 cannot be 3 because that cell sees both chain's endpoints.

What to notice: The middle numbers can change several times. Only the endpoint number needs to match.

Walkthrough

Example 02 — Check both endpoints before eliminating

This example shows another five cells chain on number 9 with starting cell r3c6 and ending cell r8c4, it eliminate number 9 from both cells r7c6 and r9c6.

What to notice: Number of cells to be eliminated can be two or more, as long as those cells see both ending cells. Cells that see only one endpoint are not covered.

How it works

THE SAME ENDPOINT NUMBER IS FORCED TO ONE END OF THE CHAIN.

Each bivalue cell contains a strong link between its two candidates. The links between cells are weak links because the cells see each other. Following the alternating implications carries the endpoint candidate from one end to the other. Since the chain starts and ends with a strong link on the same number, one endpoint must contain that number.

Start with a bivalue cell.

Choose a bivalue cell with candidate X and Y, find another bivalue cell in the same house containing either X or Y.

Pass through bivalue cells.

Move to a next cell that sees the previous cell and shares one candidate needed to continue the chain.

Return to the endpoint number.

Finish at a bivalue cell with one number either number X or Y. Remove X or Y from any cell that sees both endpoints.

CHAIN 04

A useful reminder

XY-CHAIN IS ABOUT NEIGHBORING BIVALUE CELLS SHARING A NUMBER.

The shared number between neighboring bivalue cells keep the chain moving untill the starting and ending bivalue cells of the chain share one number.

Watch & learn

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Practice

LOOK FOR PROMISING BIVALUE CELLS.

When a grid becomes rich in bivalue cells, choose an endpoint candidate and trace possible handoffs. Keep the chain short while learning, and always verify the final endpoint and elimination zone before deleting a candidate.