X-Chain
An X-Chain follows a single digit through an alternating sequence of links. A strong link joins the only two cells of a unit that can hold the digit — if one is false, the other is true. A weak link joins any two candidate cells that see each other — if one is true, the other is false. Build a chain that starts with a strong link, alternates strong and weak, and ends with a strong link; the number of links is therefore odd. The parity guarantees a powerful conclusion: at least one of the chain's two end cells holds the digit.
Why: assume the starting end is false. The first (strong) link forces its partner true; the next (weak) link forces the following cell false; and so on down the chain, alternating false-true, until the final strong link forces the far end true. Either the start is the digit, or the end is — so the digit can be removed from every cell that sees both ends.
The shortest case has three links. Take the digit 4: row 1 holds it only at r1c3 and r1c7 (strong), r1c7 sees r5c7 in the same column (weak), and row 5 holds 4 only at r5c7 and r5c2 (strong). If r1c3 is not 4, then r1c7 is, so r5c7 is not, so r5c2 is. One of r1c3 and r5c2 is 4 — and r5c3, seeing both, loses it. Skyscrapers and two-string kites are exactly these three-link chains; longer X-Chains extend the same alternation to five or seven links across the grid.
Reach for X-Chains when short single-digit patterns come up empty. Map the digit's strong links first — they are the skeleton — then look for weak connections stitching them into an odd-length chain.
- Strong links come from units with exactly two candidates for the digit; anything merely seeing each other is weak.
- The chain must begin and end on strong links, alternating throughout — an odd link count is your parity check.
- A strong link may safely play a weak role where needed, but a weak link can never stand in for a strong one.