Which of these numbers is both a perfect square AND a perfect cube?

A number is both only when every exponent in its prime factorisation is a multiple of 6, since it must be a multiple of both 2 and 3. Here $64=2^6$, so $64=8^2$ and $64=4^3$ — it qualifies. The others manage only one of the two: $144=12^2$ is a square but ∛144 ≈ 5.24, while $216=6^3$ and $512=2^9=8^3$ are cubes but √216 ≈ 14.7 and √512 ≈ 22.6. The smallest numbers that are both are 1, 64, 729 and 4096 — the sixth powers.