TITLE: an extension of an Olympiad problem
QUESTION [6 upvotes]: I'm a high school student and I'm currently working on my math extended essay. I chose an Olympiad question which is:
A piece of land of a square shape with dimensions 10m x 10m is divided into 100 square parcels with dimensions 1m x 1m. Initially, 9 of the parcels are overgrown by weed. If a parcel is surrounded by at least 2 parcels with weed from its sides after some time that parcel will be overgrown by weed. Can the whole piece of land grow into weed after some time?
I already have a rough idea on how to solve the original question, but my essay involves an extension to 3D, namely like this:
A space of land of a cube shape with dimensions 10m x 10m x 10m is divided into 1000 cubes with dimensions 1m x 1m x 1m. What is the minimum numbers of cubes to begin with so that the whole cube would grow into weed after some time, given that if a cube is surrounded by at least two other cubes with weed it will also be overgrown by weed?
After some thinking I have a hypothesis (which I have no idea whether it's correct or not). My approach start with 2D, for it is proven in the first part (somehow) that for a n x n square to grow into weed, n parcels are needed to begin with. Simplifying to a 3 x 3 x 3 cube, that means at least 3 cubes are needed for one side to be overgrown. Let the side be (x, y, z1). Now just a random cube in layer (x, y, z3) can lead to the whole cube being overgrown. I also made an expression for this:
c = n+z*, where z* means n/2 rounding backwards (for example 2.5 to 2) and c is the number of cubes needed. It's a bit messy but well.
The problem is, I don't know how to prove this hypothesis mathematically. The proving methods I learned like induction, direct proof etc. doesn't really apply here, which is why I'm stuck. Could you maybe point out some directions for me so that I can do some research and try proving it? Or, if it is impossible to prove, please also inform me so that I can change my plan. If it's solvable (which to me isn't right now), please don't post the solution because if so I would have to find another topic to do for my essay. I just want to know some proving methods which might be helpful in this case, thank you!

REPLY [4 votes]: Here are some hints:
Project the $n\times n\times n$ cubes into $n\times n$ squares, must the diagonal in the projection fully overgrown by weed? What does it tell us about the minimum number or overgrown cubes in the beginning?
In your example, projection of first layer and third layer overlap by exactly one square. Must the projection between consecutive odd layers overlap (answer is no, but in the even layer between them there must be ...)? What does it tell you about the number of cubes?