I and J are sisters. J is married to K. L is K’s father. M is the wife of L. N…

Concept: A blood-relation chain is read as a family tree assembled one link at a time, and only two kinds of link occur in it. A blood link (parent–child or sibling) fixes a person's generation and the family line on which that person sits, while a marital link (husband–wife) joins two people of the same generation and lets each of them borrow the other's blood relatives. By the standard convention of these puzzles, every link is read in its simplest form unless the question says otherwise: the spouse of a person's parent is read as that person's other parent, and children of the same parents are read as full siblings. A relation is named an in-law relation exactly when the shortest path between the two people crosses a marital link.

Application: take the statements in the order given and add one link at a time.

1. I and J are sisters — I and J are female siblings standing in the same generation.

2. J is married to K — K is J's husband, and this marital link is the only route by which J is connected to the rest of the family.

3. L is K's father — L stands one generation above K, on K's own blood line.

4. M is the wife of L — M is the wife of K's father and, by the convention stated above, is read as K's mother.

5. N is M's daughter — N is a daughter of K's mother, so N and K are read as siblings; N therefore stands in K's own generation.

6. N reaches J only through K: N is K's sister and K is J's husband, so N is the sister of the man J married.

Cross-check: the path from J to N crosses exactly one marital link (J–K) and then one sibling link (K–N), so N is a same-generation relative of J on the husband's side. Reading the chain from the other end confirms it: N's brother K is the man J married.

• Sister would require N and J to be siblings on a shared parental line; N's line here runs through L and M, whereas J's own parents are never mentioned — J enters this family only by marrying K.

• Father's sister would place N one generation above J on J's paternal line; N is instead in J's own generation, being a sibling of J's husband.

• Mother's mother would place N two generations above J on the maternal line; the mother in this chain is M, and she is K's mother, not J's.

Hence N is the sister of J's husband — N is J's husband's sister.