If A # B means A is father of B; A $ B means A is mother of B, A @ B means A…

Concept: In this puzzle's coding scheme, each of the three defined codes (A#B = A is father of B; A$B = A is mother of B; A@B = A is sister of B) fixes a specific family role and directly states the gender of the person named FIRST in that code (the father, mother, or sister); none of these three codes, as defined here, states the gender of the person named SECOND. So in a chain built purely from these codes, a person's own gender is confirmed only when some code in the chain actually places them as the first name — not merely as someone reached through another person's parent, sibling, or child code.

1. X $ K: X is the mother of K, so X is one generation above K.

2. K # A: K is the father of A, so K is one generation above A, making X two generations above A (X is A's grandmother).

3. A @ B: A is the sister of B, so A and B share the same parents; since K is A's father, K is also B's father, and since X is K's mother, X is B's grandmother through the same line.

This chain fixes X exactly two generations above B and settles that the link runs through the shared father K, but it does so using only the mother-code (on X), the father-code (on K), and the sister-code (on A) — none of the three codes is ever applied with B as the first-named person, so B's own gender is never stated anywhere in the expression.

Cross-check: Testing each of the three given codes in turn confirms this — the mother-code names X, the father-code names K, and the sister-code names A; B appears only as the second name in the last code, the position whose gender a code never fixes. So the generation gap (grandparent-grandchild) is certain, while the specific title depends on information the puzzle never supplies.

• Grandson — assumes B is male, but no code in the expression is ever applied with B as the person whose own gender it fixes, so this specific gender cannot be confirmed.

• Grand-daughter — assumes B is female for the same unconfirmed reason; the sister-code fixes A's gender, not B's.

• Nephew — this would require X to belong to the same generation as K's siblings, but the mother-code places X a full generation above K, not alongside K, so a same-generation title does not fit the chain at all.

• Data Inadequate — the chain fixes X two generations above B beyond doubt, but since none of the three given codes ever states B's own gender, no single specific relationship term can be selected between the two grandparent-grandchild options; this is the only choice consistent with exactly what the codes establish.

So B is confirmed to be X's grandchild through the shared father K, but whether B is a grandson or a granddaughter cannot be fixed from the coded expression — which is why Data Inadequate is the correct choice.