How many three-digit perfect cubes are there whose sum of the digits is a perfect square?

There are 4 three-digit perfect cubes whose sum of the digits is a perfect square. These numbers are 125, 512, 729, and 926.

Solution:

We can list out the three-digit perfect cubes that exist:

- 125
- 216
- 343
- 512
- 729

Now, we need to find those whose sum of the digits is a perfect square.

- 125: 1 + 2 + 5 = 8, not a perfect square
- 216: 2 + 1 + 6 = 9, a perfect square
- 343: 3 + 4 + 3 = 10, not a perfect square
- 512: 5 + 1 + 2 = 8, not a perfect square
- 729: 7 + 2 + 9 = 18, not a perfect square

Therefore, there is one three-digit perfect cube whose sum of the digits is a perfect square: 216.

Answer: 1