TITLE: How can I calculate the number of perfect cubes among the first $4000$ positive integers?
QUESTION [11 upvotes]: How can I calculate the number of perfect cubes among the first $4000$ positive integers?
Is there any trick to solving such questions?

REPLY [22 votes]: If you live in the world of base-2 geekdom, simply note that $2^{12}$ a.k.a. $16^3$ is 4K, or 4096.  This is obviously too large.  The barest mental math estimation will verify that $15^3 < 4000$.  Done.