Advanced technique · Chains · 02

X-Chain

One candidate travels through alternating strong and weak links.

AdvancedChainsElimination

The idea

ONE NUMBER. ALTERNATING LINKS. TWO STRONG ENDPOINTS.

An X-Chain uses only one candidate number. The chain alternates strong and weak links between occurrences of that number. The important condition is that the chain starts and ends with a strong link, which constraints a valid X-Chain to must have an odd number N of links connecting an even number N+1 of cells, and N is at least 3.

That structure proves that at least one endpoint contains the chain number. Any cell that sees both endpoints therefore cannot contain that number.

The question: Can one candidate travel through alternating strong and weak links and finish with strong links at both ends?

Walkthroughs

TRACE ONE NUMBER THROUGH THE GRID

Focus on one number and ignore others while tracing an X-Chain. At each step, label whether the connection is strong or weak, and make sure the link types alternate. Remember a strong link can act as a weak link, but a weak link can never act as a strong link.

Walkthrough

Example 01 — A typical single-number chain

For number 5, a chain starts from cell r6c5 and travels through cells r6c6, r7c6, r7c3, r1c3, r3c1, r3c8 and ends at r2c8. The chain contains seven links and eight cells.

Because the chain begins with a strong link in row 6 and ends with a strong link in column 8, one of cells r6c5 and r2c8 must be number 5. Any cell that sees both endpoints cannot be number 5.

What to notice: The chain does not need to look like a regular shape. The proof comes from the alternating link structure.

Walkthrough

Example 02 — Check the elimination target

For number 2, a chain of seven links and eight cells starting from cell r1c1 and ending at cell r9c9, both starting and ending links are strong, therefore, number 5 can be removed from cell r9c1 because it sees both starting and ending points.

What to notice: Strong endpoints are what turn the chain into an elimination.

After finding a valid X-Chain, do not eliminate the number from every cell near the chain. Identify cells that see both strong endpoints. Those cells are the only guaranteed elimination zone for the endpoint conclusion.

A candidate that sees only one endpoint is not covered by the same proof. The two-endpoint relationship is the key.

How it works

STRONG ENDPOINTS PROVE THAT ONE END MUST CONTAIN THE CHAIN NUMBER.

Read the chain as an alternating sequence. If the first endpoint is false, its strong link forces the next candidate true; the weak link then forces the next one false, and so on. Because the final link is strong, the opposite endpoint becomes true. Reversing the chain gives the same conclusion from the other direction: at least one endpoint is the number.

Choose one number.

Pick a candidate with useful strong links. Ignore other numbers while building this chain.

Alternate the links.

Move through strong and weak links in alternating order. A strong link may also serve as a weak link, but not vice versa.

Use the endpoints.

Because both ends are strong, eliminate the number from any cell that sees both endpoints.

CHAIN 02

A useful reminder

THE FIRST AND LAST LINKS MUST BE STRONG.

If the chain does not begin and end with strong links on the same number, you do not have the standard X-Chain elimination described here. Check the link types before removing anything.

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Practice

TRACE ONE NUMBER AT A TIME.

Choose a candidate with several conjugate pairs. Mark the strong links first, then connect them with weak links and look for a chain that finishes on a strong link. Keep the elimination zone limited to cells that see both endpoints.