Nine cells of a $10 \times 10$ diagram are infected. At each step, a cell becomes infected if it was already infected or if it had at least two infected neighbors (among the 4 adjacent cells).

(a) Can the infection spread everywhere?

(b) How many initially infected cells are needed to spread the infection everywhere?
(a) No, this is not possible. Note that the perimeter of the area delimited by the infected cells decreases in a broad sense at each step. Initially, it is at most $9 \times 4=36$, and cannot reach 40.

(b) According to question (a), at least 10 cells must already be infected. We place 10 cells along a diagonal. We verify that then the infection can spread everywhere.