Dashboard infection

1. It is not enough to count infected; you need a geometric magnitude that cannot grow without limit under that rule.
2. Look at what happens to the perimeter or the orthogonal border of the infected set.
3. A new cell requires at least two infected neighbors, so the expansion cannot fill the entire board from the initial 99.

Answer: No. It is impossible to infect all 10,000 cells. Defines $P$ as the number of edges between infected and non-infected squares (including the outer edge of the board). When a healthy square becomes infected, it had $k\ge2$ infected orthogonal neighbors. - $k$ border edges are removed,

• $4-k$ new borders are added. Then the net change is:

$$ \Delta P=(4-k)-k=4-2k\le0. $$

So $P$ never increases. At the beginning, with 99 infected squares:

$$ P_0\le 99\cdot4=396. $$ If the entire $100\times100$ board were infected, the outer border would be: $$

P_f=4\cdot100=400.
$$ But that would require growing from $\le396$ to 400, a contradiction. Conclusion: total infection cannot occur.