Forcing Y-Chain
A Forcing Y-Chain is an open forcing chain built entirely from bivalue-cell links: it never uses a single-digit strong or weak link. Each step passes through a cell with exactly two candidates — if one candidate is true, the other must be false, and vice versa. Because every link is a bivalue-cell link, the chain is free to switch digits at every step, unlike a Forcing X-Chain, which stays on one digit throughout. The chain starts at an assumption in a bivalue cell and follows a sequence of bivalue cells to a conclusion; if that conclusion contradicts the assumption, or forces a value elsewhere, an elimination follows.
What is a Forcing Y-Chain?
A Forcing Y-Chain is an open chain that uses only bivalue-cell links (Y-links) — it never relies on a strong or weak link tied to a single digit. A Y-link passes through a cell with exactly two remaining candidates: if one candidate is assumed true, the other must be false, because the cell can only hold one value; conversely, if one candidate is false and it is the only alternative left in that cell, the other must be true. The chain starts at a cell where you assume a candidate is true (or false) and follows a sequence of bivalue cells, at each step flipping to the other candidate in that cell, which may belong to a different digit than the one before. This is what distinguishes a Forcing Y-Chain from a Forcing X-Chain: an X-Chain tracks a single digit across many houses using strong and weak links, while a Y-Chain tracks a sequence of bivalue cells and can hop between different digits at every step. The chain continues until it reaches a conclusion — a contradiction with the starting assumption, or a forced value that allows an elimination outside the chain.
How to spot it
To find a Forcing Y-Chain, first scan the grid for bivalue cells — cells with exactly two pencil-mark candidates. Pick a candidate in one of these cells and assume it is true; the other candidate in that cell must then be false. If that newly false candidate shares a bivalue cell with another digit, the second candidate in that cell must be true, and the chain continues into a fresh bivalue cell. Trace this sequence of bivalue cells, remembering which candidate is asserted true or false at each stop, and note whether the chain switches digits along the way — a sign the chain is a genuine Y-Chain rather than a disguised X-Chain, which stays on one digit throughout. Look for chains that end either in a contradiction with the starting assumption, or with a candidate proven true or false such that a peer cell can have that candidate eliminated. Since Forcing Y-Chains rely purely on bivalue cells, puzzles with plenty of two-candidate cells are the easiest places to find them.
The logic
A Forcing Y-Chain works by propagating a truth value through a sequence of bivalue cells rather than through strong and weak links on a single digit. In a bivalue cell with candidates {A, B}, the two candidates are logically linked: exactly one of them must be true, so asserting A is true forces B false in that same cell, and asserting A is false forces B true. The chain starts with an assumption about one candidate, then moves from bivalue cell to bivalue cell, at each step turning the newly forced value in one cell into an assumption about a shared candidate in the next bivalue cell — the same digit appearing in both cells, or the two cells sharing a house on that digit. Because the digit can change at every hop, the chain is more flexible than a single-digit X-Chain, but it depends entirely on the presence of two-candidate cells to advance. If the chain reaches a cell where the original assumption forces a contradiction, the assumption was wrong and the candidate can be eliminated at the start. If instead the chain proves a candidate must be true, any other cell that sees both the starting cell and the concluding cell for that candidate cannot hold it, and the elimination follows.