Dashboard infection

1. When a healthy square became infected, it had kge2 infected orthogonal neighbors.
2. Then the net change is: Delta P=(4-k)-k=4-2kle0. So P never increases.
3. If the entire 100×100 board were infected, the outer border would be: P_f=4·100=400.

Back to problem
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,
• new $4-k$ 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.