Code: 64 to 343, 216 to 729, find 512

Identify the mathematical nature of the numbers provided. All of them are perfect cubes. 64 = 4³ is coded as 343 = 7³. The base relationship is 4 \rightarrow 7 (an increase of +3). 216 = 6³ is coded as 729 = 9³. The base relationship is 6 \rightarrow 9 (an increase of +3). The code operates by taking the cube root, adding 3, and cubing the result. Apply this to the target number 512. 512 = 8³. Adding 3 to the base gives 8 + 3 = 11. Calculate the cube of 11: 11³ = 1331.

Answer: (b).