A bacterium develops on a $100 \times 100$ grid. It can infect a new cell if and only if two adjacent cells were already infected. What is the minimum number of initially infected cells required for the bacterium to spread everywhere?
We observe that the perimeter is monovariant: it can only decrease. The initial perimeter is at most 4 times the number of initial cells. The final perimeter is $4 \times 100$. Therefore, it requires at least 100 cells, and we can take the cells of a diagonal.