Who is Jill's uncle? Statements: I) Jill's father Bill's only nephew Jack is…

Concept: In a blood-relation data-sufficiency problem, a statement is sufficient only when it pins down a unique person of the required gender. Two tools matter here: a plain "X is Y's son" fixes only the child's gender — not the parent's, and not whether that parent is the blood relative in the chain or merely that relative's spouse; a gendered marriage term such as "brother-in-law" is inherently male, and when it is qualified by "only" (a uniqueness condition) it can pin a single person even though the phrase itself has more than one meaning.

Application — Statement I: Bill is Jill's father, so Bill's only nephew Jack is Jill's only paternal cousin, and Jack is Sam's son. "Son" leaves Sam's own gender open and does not say whether Sam is Bill's sibling or that sibling's spouse. So the uncle could be Sam (if Sam is a male brother or brother-in-law of Bill) or an unnamed person (if Sam is female, i.e. an aunt). No unique uncle, so Statement I alone is not sufficient.

Application — Statement II: This only places Frank as the maternal aunt's sole brother-in-law; on its own it fixes neither Jill's father nor the paternal-side structure needed to tie Frank to Jill as an uncle, so Statement II alone is not sufficient.

Application — Both together: Combine them (Jack is Jill's only paternal cousin, and Jack's maternal aunt is a sister of Jack's mother):

1. Jack's father is male and is the husband of Jack's mother, who is a sister of the maternal aunt. So Jack's father is necessarily one of that aunt's brothers-in-law (the husband of her sister) — this holds on either reading of "brother-in-law," so the phrase's ambiguity does not matter.

2. Statement II says the aunt has only ONE brother-in-law, Frank. Since Jack's father is always such a brother-in-law, that single brother-in-law must be Jack's father himself; hence Frank is Jack's father. (Frank and Sam are distinct names, so this also rules out the readings in which Sam were the uncle.)

3. Jack's father is Jill's uncle whatever the family shape: if Bill's sibling on this chain is his brother, that brother is Jack's father and Jill's paternal uncle; if the sibling is Bill's sister, her husband is Jack's father and Jill's uncle by marriage. Either way Jill's uncle = Jack's father = Frank, a single answer.

Cross-check: Neither statement alone identifies the uncle, but combined they force a unique answer — Frank. Therefore both statements together are sufficient.