In a certain code language,P + Q means 'P is the son of Q', P & Q means 'P is…

Concept: in a symbol-coded blood-relation chain every operator fixes two things at once — the relationship it names and the gender of the person written on its left. '&' makes that person a husband (male), '%' makes that person a mother (female), '#' makes that person a sister (female), and '+' makes that person a son (male).

A chain is read link by link from left to right; each link binds only the pair it stands between, and the links together assemble one family tree. By definition X is the father-in-law of Y when X is male and Y is married to a child of X, so a chain qualifies only when it makes F male, gives F a child, and marries that child to M.

Applying the definition to each chain:

• Chain I — F & T % S # U & M

  1. F & T: F is the husband of T, so F is male and T is his wife.

  2. T % S: T is the mother of S, so S is a child of T and of her husband F.

  3. S # U: S is the sister of U, so U is a sibling of S and therefore another child of F and T.

  4. U & M: U is the husband of M, so M is the wife of U.

  M is married to U, who is a child of F, and F is male — every part of the definition is met.

• Chain II — F # T % S # U & M

  1. F # T: F is the sister of T, so F is female.

  2. T % S: T is the mother of S.

  3. S # U: S is the sister of U, so U is another child of T.

  4. U & M: U is the husband of M.

  F is the sister of the mother of M's husband, so F is an aunt of M's husband, and a female relative cannot be a father-in-law.

• Chain III — F & T % S # U % M

  1. F & T: F is the husband of T.

  2. T % S: T is the mother of S, so S is a child of F and T.

  3. S # U: S is the sister of U, so U is another child of F and T.

  4. U % M: U is the mother of M, so M is the child of U.

  M is the child of a child of F, so F stands as M's maternal grandfather rather than as the father of M's spouse.

• Chain IV — F & T # S # U % M

  1. F & T: F is the husband of T.

  2. T # S: T is the sister of S, so S is T's sibling and no parent-child link is created here.

  3. S # U: S is the sister of U, so U is a sibling of T as well.

  4. U % M: U is the mother of M.

  F is the husband of the sister of M's mother, so F stands as M's uncle by marriage.

Cross-check — the relation each chain assigns to F with respect to M:

Exactly one chain, I, satisfies the definition of father-in-law, so the intended answer is 'Only I'.