There are 100 closed lockers in a hallway. 100 cats walk by. The first cat toggles every locker. The second cat toggles every 2nd locker. The third cat toggles every 3rd locker. And so on — the nth cat toggles every nth locker. After all 100 cats have passed: (a) which lockers are open? (b) Why? Prove your answer mathematically.

Judge: Correct identification of perfect squares as the answer with all 10 lockers listed. Provides a rigorous mathematical proof: connects toggling to divisor count, proves divisors come in pairs except for perfect squares, and formally shows why only squares have odd divisor counts. Both informal and formal proof versions are provided. Exceptionally clear.

Judge: Complete response with the correct answer, all 10 lockers, and a full pairing proof culminating in d^2 = m. Clear and self-contained.

Judge: Correctly identifies all 10 perfect-square lockers and gives a rigorous proof: toggles equal divisor count, divisors pair off except when d = k/d (perfect squares), so only squares have an odd count. Both hard constraints met. Exceptionally clear.

Judge: Identifies all perfect squares and proves the result via the divisor-pairing argument with the square-root self-pairing exception. Mathematical explanation is complete and both hard constraints are satisfied. Clear and concise.

Judge: Correctly lists all 10 open lockers (perfect squares 1-100) and gives a rigorous proof: locker n is toggled once per divisor, divisors pair as (d, n/d) except when d^2 = n, so only perfect squares have an odd divisor count. Includes worked examples (12 and 36) that reinforce the argument. Both hard constraints are clearly met.

Judge: Correctly lists all 10 perfect-square lockers and gives a rigorous divisor-pairing proof (d paired with k/d, self-paired only when d^2 = k), plus the prime-factorization formulation as a second proof. Worked examples (12 and 36) are arithmetically correct. Exemplary on every dimension.

Judge: Correct answer (perfect squares 1-100) with a rigorous mathematical proof. The proof uses prime factorization to show d(n) is odd iff all exponents are even iff n is a perfect square. This is a textbook-quality proof, not just an intuition. Lists all 10 open lockers. The divisor function formulation is clean and the converse direction is explicitly addressed. Excellent mathematical writing.

Judge: Correct identification of perfect squares (1,4,9,...,100) with a rigorous mathematical proof. Clearly establishes that locker k is toggled tau(k) times, that tau(k) is odd iff k is a perfect square, and proves the pairing argument for divisors. Lists all 10 open lockers. The proof is clean, well-structured, and mathematically complete.

Judge: Correctly identifies perfect squares as the answer and lists all 10. The mathematical proof clearly explains divisor pairing and why perfect squares have an odd divisor count, using 36 as a concrete example. Solid proof though could be slightly more formal.

Judge: Correctly identifies perfect squares (1,4,9,16,25,36,49,64,81,100) as the answer. Provides the key insight about divisors coming in pairs except when d=n/d (perfect squares). Begins a formal proof using prime factorization but the response is truncated. The divisor-pairing argument is the correct mathematical explanation. Lists all 10 open lockers. The truncation prevents seeing the complete proof, but the visible reasoning is rigorous.

Judge: Correctly identifies that perfect squares are the open lockers. The proof is solid: divisors come in pairs (d, n/d), except when d = sqrt(n), so only perfect squares have an odd number of divisors. Uses concrete examples (12 and 16) to illustrate. The response is truncated before listing all 10 perfect squares, but the mathematical reasoning is complete and rigorous.

Judge: Correct answer (perfect squares) with all 10 lockers listed. The proof is well-structured: establishes divisor-toggle connection, explains why divisors pair up, shows perfect squares are the exception. Truncated at the end but the core proof is complete. Uses good concrete examples (12 vs 16).

Judge: Correctly identifies perfect squares as the answer with a sound mathematical proof. Explains that divisors come in pairs except for square roots, giving perfect squares an odd number of divisors. The response is truncated before listing all 10 perfect squares, but the reasoning and proof are solid. Covers both the 'what' and 'why' as requested.

Judge: Completely wrong answer. Claims the open lockers are primes, when the correct answer is perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100). The reasoning is fundamentally flawed -- incorrectly claims a locker is toggled only if its number is prime, which contradicts basic number theory (a locker is toggled once for each of its divisors). Fails both hard constraints.