2. On a $2019 \times 2019$ chessboard, a contagious disease is spreading. Each day, some of the squares on the chessboard are sick, and the rest are healthy. A healthy square that borders (along a side) a sick square becomes sick the next day. A sick square is always healthy again the next day. A healthy square that has been sick before can become sick again (if it is infected by an adjacent sick square). On day 1, only the middle square is sick. How many squares are sick on day 100?
A) 200
B) 298
C) 396
D) 9999
E) 10000
A2. E) 10000 If we look at which cells are sick on day 1, 2, 3, and 4, we see that these cells form a perfect square rotated by $45^{\circ}$ (see below). On day 1, there is $1^{2}=1$ cell sick, on day 2 there are $2^{2}=4$, then $3^{2}=9$, and then $4^{2}=16$. On day 5, the same cells are sick as on day 3, plus a whole new layer of sick cells around it, making it a $5 \times 5$ square. This pattern continues, and on day 100, there are thus $100^{2}=10000$ cells sick.

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-1.jpg?height=362&width=354&top_left_y=1144&top_left_x=268)

day 1

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-1.jpg?height=362&width=369&top_left_y=1144&top_left_x=655)

day 2

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-1.jpg?height=362&width=368&top_left_y=1144&top_left_x=1044)

day 3

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-1.jpg?height=362&width=366&top_left_y=1144&top_left_x=1439)

day 4