A chain is a sequence of logically connected cells used in advanced solving techniques. Each link follows from the last, letting you trace implications across the grid until a contradiction or forced placement appears.
A chain is an ordered sequence of cells where each cell is linked to the next through a shared candidate. Chains are the backbone of advanced Sudoku, letting you reason across the grid instead of one house at a time.
Two cells are connected by a strong link when a candidate has only two possible cells in their shared house - if it's not one, it must be the other. They are connected by a weak link when they merely share a house and a candidate. Real chains alternate between strong and weak links.
You will encounter chains in Simple Coloring, Forcing Chains, and XY-Wing (a 3-cell chain). Mastering them is what separates intermediate solvers from advanced ones.
The power of a chain comes from its end-points. If you can show that two cells at opposite ends of a chain see (share a house with) a third cell containing the chained candidate, that candidate can be eliminated from the third cell - regardless of which end of the chain is "true."
The proof works by case analysis: assume the start of the chain is true, then false. In one case, one end-point places the digit; in the other case, the opposite end places it. Either way, the cell that sees both ends cannot also hold that digit.
The shortest useful chain is the XY-Wing, a three-cell chain where each cell holds exactly two candidates. Simple Coloring chains alternate two colors along strong links for a single digit.
Forcing Chains generalize the idea: you assume a candidate is placed, follow every implication, and look for a contradiction or a cell that gets the same value down both branches. See the full glossary for related terms like conjugate pair and candidate.
A strong link exists when a candidate has exactly two possible cells in a house - if one is false, the other must be true. A weak link only requires that two cells share a house and a candidate; both could be false. Real chains alternate strong and weak links.
No. Easy and medium puzzles solve with singles, pairs, and locked candidates. Chains start to matter at hard difficulty and become essential for expert and extreme puzzles where simpler techniques run out.
The XY-Wing - a three-cell chain where every cell has exactly two candidates - is the gentlest entry point. From there, Simple Coloring and X-Chains add length without adding much new theory.
Pick a starting cell with two candidates. Here R5C5 has [3, 8]. We'll test BOTH branches and watch where they lead.
Step 1 of 4 — full interactive demo available with JavaScript enabled
Ready to put your skills to the test?
Try Final Sudoku - an ad-free Sudoku experience with smart tools