Two-Node X-Chain (Empty Rectangle - minimal)
A Two-Node X-Chain — traditionally listed as 2 Strong Links — is the general length-2 same-digit AIC. Its signature is (A=B)-(C=D): two strong-inference nodes joined by one weak inference. The chain proves the inclusive endpoint relation A OR D, so the digit can be removed from any candidate that sees both endpoints.
What is a Two-Node X-Chain?
Write the construction as (A=B)-(C=D). Each equals sign is a strong link (XOR node): at least one side is true, and for an ordinary two-position house node the sides are also mutually exclusive. The dash is a weak inference (NAND), meaning B and C cannot both be true. The node houses may be any combination of rows, columns, and boxes; the logical relations, not a bridge drawing, define the chain.
Walkthrough example
A walkthrough of one Two-Node X-Chain written as (A=B)-(C=D), with two strong inferences, one weak inference, and a common-peer elimination from the endpoints.
Step 1
Consider the two-node X-Chain for 2 in row 4 and block 7.
Step 2
Write the chain as (G4=C4)-(C8=B7). Each equals sign is a strong link (XOR): in row 4, exactly one of G4 and C4 contains 2; in block 7, exactly one of C8 and B7 contains 2. The dash is a weak inference (NAND): C4 and C8 see each other in this column, so they cannot both contain 2. If G4 is false, C4 is true, C8 is false, and B7 is true. Therefore the endpoints satisfy G4 OR B7: at least one contains 2, possibly both.
Step 3
This is the Type 1 same-digit elimination: because G4 OR B7 is proved for 2, remove 2 from every candidate that sees both endpoints.
The signature (A=B)-(C=D) supplies two strong links (XOR) and one weak inference (NAND), proving the inclusive endpoint relation A OR D. That relation justifies removing the digit from a candidate that sees both endpoints. This is the Type 1 same-digit elimination. You can play free Sudoku to practise with the current candidate state visible.
How to spot it
For one digit, find two justified strong-inference nodes and connect one side of each with a valid weak inference. The unconnected sides A and D are the endpoints. A target is valid only when it conflicts with both endpoints. Skyscraper and 2-String Kite are named house-signature restrictions of this general two-node X-Chain.
The logic
If A is false, the first strong link makes B true. The weak inference makes C false, and the second strong link makes D true. Thus A OR D is true: at least one endpoint contains the digit, possibly both. Any candidate that sees both endpoints would conflict whichever endpoint is true and can therefore be removed. This is the Type 1 same-digit elimination.