This problem has been brought up here a couple of times so I'm doing a copy and paste for the introduction:
“There is a dangerous bacteria, Checkerichia Coli, (or C. Coli for short) which lives in the squares of checkerboards. Squares with this bacteria living in their digestive systems are said to have "C-sickness". Once a square has "C-sickness," it is stuck with it forever!
Excessive contact with C-sick squares will cause a square to become C-sick. Specifically, if a healthy square has at least two C-sick neighbors (orthognal, not diagonal), then that square becomes C-sick the next day.
For example, on the left we the photo of a see a partially infected checkerboard which was taken yesterday. On the right is a photo taken today.” enter image description here
The original question under this problem was "If the terrorist has only 7 samples, can he still infect the whole board?“ which has been proven by comparing the maximal parameter of each case and thus finding out the minimal number of initially infected squares required to eventually infect the entire board. Is there a way to generalise this problem(proving this mathematically, finding out a formula for the process, etc)? Could we then extend this generalisation to a 3D cube(the infection rule could tweak a little, for instance requiring a block to border 3 instead of 2 already infected blocks in order to be infected) and find out the minimal number of initially infected blocks required in order to eventually infect the entire cube?
Context: this is a task given by my teacher in high school. My tries: I couldn't figure out a mathematical proof but have proved through exhaustion that the minimal initially infected blocks needed to eventually infect the entire cube is $2n-1$ if a blocks gets infected when bordering two already infected blocks, and $n^2$ if it needs to border three. I haven't studied discrete mathematics so inputs on this regard(proof by case analysis) would be greatly appreciated as well:)
Original post: https://puzzling.stackexchange.com/questions/18074/checkerboard-infection