2018 AMC 8

Solution 1

We compute . We're all familiar with what is, namely , which is too small. The smallest cube greater than it is . is too large to calculate, but we notice that , which therefore will clearly be the largest cube less than . So, the required number of cubes is .

Solution 2

First, . Then, . Now, we can see how many perfect cubes are between these two parameters. By guessing and checking, we find that it starts from and ends with . Now, by counting how many numbers are between these, we find the answer to be .