Generalized X-Wing (X-Cycle)
A Generalized X-Wing is a four-cell single-digit cycle — the smallest X-Cycle in Sudoku. Pick a candidate digit and follow a chain of strong and weak links that loops back to its starting cell. When the loop passes through exactly four cells, the pattern behaves like an X-Wing: the digit is locked into two opposite corners, and any candidate visible from both sides of the cycle can be eliminated.
What is a Generalized X-Wing?
A Generalized X-Wing is the simplest closed X-Cycle — a single-digit forcing chain that returns to its origin after visiting just four cells. The chain alternates between strong links (a digit confined to two cells in a house) and weak links (a digit appearing in multiple cells but only one can be true). When the loop closes at four cells, the logic mirrors a classic X-Wing: two cells must be true and two must be false, arranged so that each row and column in the cycle gets exactly one placement. Any cell outside the cycle that sees both a true node and a false node for the same digit cannot hold that digit, so it gets eliminated. The difference from a regular X-Wing is that the four cells need not form a neat rectangle — they can be scattered across different rows, columns, and boxes, connected by the chain's alternating links.
How to spot it
Generalized X-Wings require thinking in chains, not just scanning rows and columns. Start by picking a candidate digit and tracing strong links — houses where that digit can only go in two cells. If you can connect two strong links through a shared cell or a weak link, and the chain closes back on itself after four cells, you have a Generalized X-Wing. The eliminations fall on cells that are visible from two nodes of opposite polarity in the cycle — one where the digit is on and one where it is off. These patterns are rare and typically appear only after all singles, subsets, and fish patterns have been exhausted.
The logic
An X-Cycle is a chain of alternating strong and weak links on a single digit. A strong link means one of these two cells must contain the digit; a weak link means at most one of these cells can contain the digit. When the chain forms a closed loop of four cells, the alternating pattern forces two cells to be true and two to be false. Any cell outside the loop that sees both a true node and a false node for the same digit faces a contradiction: if the digit is placed there, one side of the cycle breaks. So that candidate can be removed. The four-cell case is called a Generalized X-Wing because it produces the same elimination pattern as a regular X-Wing, but without requiring the four cells to align in a perfect rectangle.