Bidirectional X-Cycle
A Bidirectional X-Cycle is a closed single-digit forcing-chain loop with six or more cells. The chain alternates strong and weak links on one candidate digit, and because the loop closes, every node can be evaluated in both directions — forward and backward — which confirms the polarity of each cell. Any candidate outside the loop that sees both a true node and a false node for the same digit can be eliminated.
What is a Bidirectional X-Cycle?
A Bidirectional X-Cycle is a closed X-Cycle with more than four cells — the four-cell case is called a Generalized X-Wing. The chain uses a single candidate digit and 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, the alternating pattern assigns a polarity — true or false — to every node. The term bidirectional means each node's polarity can be verified by traversing the loop in either direction, which makes the deduction especially robust. Any cell outside the loop that sees both a true node and a false node for the same digit cannot hold that digit, so it gets eliminated. Longer cycles — six, eight, ten cells or more — produce the same kind of eliminations as the four-cell Generalized X-Wing, but on a wider scale.
How to spot it
Bidirectional X-Cycles require tracing chains of strong and weak links on a single digit. Start by picking a candidate digit and identifying strong links — houses where that digit can only go in two cells. Connect strong links through shared cells or weak links, and check whether the chain closes back on itself. If the loop has more than four cells and alternates correctly, you have a Bidirectional X-Cycle. The eliminations fall on cells that are visible from two nodes of opposite polarity — one where the digit is on and one where it is off. These patterns are rare and typically appear only after all singles, subsets, fish, and simpler chain techniques 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 two cells must contain the digit; a weak link means at most one of them can. When the chain forms a closed loop, the alternating pattern forces each node to a definite polarity — true or false. Because the loop is closed, you can confirm each node's polarity by traversing in either direction, hence bidirectional. Any cell outside the loop that sees both a true node and a false node for the same digit faces a contradiction: placing the digit there would break one side of the cycle. So that candidate can be removed. The four-cell case is a Generalized X-Wing; longer loops are Bidirectional X-Cycles, and both produce the same type of elimination.