You may remember the C. Coli bacteria, which infects checkerboard squares. There is a similar disease which plagues cubes, known as cubic lice. Once a cube gets infested with cubic lice, it will be infested for the rest of its life!
A refresher on cube biology: cubes live together in colonies, which consist of $1000$ cubes tightly packed together in a $10\times 10 \times 10$ cubical pattern. If a healthy cube is ever adjacent to at least three infested cubes, then that cube will itself become infested the next day.
A bioterrorist wants to infest an entire colony of cubes with cubic lice. He plans to break into a WHO lab and steal several cubic larvae, each of which can cause a single cube to become infested.
What is fewest number of larvae he needs to steal to be able to infect the entire colony?
As usual, you should prove your answer is correct, by showing how he can infect the colony with that many larvae, and why he can't with fewer.