On a $123 \times 123$ board, each cell is painted purple or blue according to the following conditions:

- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.
- Each blue cell that is not on the edge of the board has exactly 4 purple cells among its 8 neighbors.

Note: Two cells are neighbors if they share a side or a vertex.

(a) Consider a $3 \times 3$ board within the $123 \times 123$ board. How many cells of each color can there be in this $3 \times 3$ board?

(b) Calculate the number of purple cells on the $123 \times 123$ board.
(a) Observing a $3 \times 3$ board, we can clearly see that its center is not on the edge of the board. The center cell can:

- Be painted purple. In this case, among its 8 neighbors, there are 5 blue and 3 purple. In total, there are 4 purple cells and 5 blue cells on this board.
- Be painted blue. In this case, among its 8 neighbors, there are 4 blue and 4 purple. In total, there are 4 purple cells and 5 blue cells on this board.

(b) Since in any $3 \times 3$ board within the $123 \times 123$ board, the number of blue cells is 5 and the number of purple cells is 4, we can divide the $123 \times 123$ board into smaller $3 \times 3$ boards as shown in Figure 69.1. In this way, the board is divided into $\left(\frac{123}{3}\right)^{2} = 41^{2} = 1681$ $3 \times 3$ boards. Since each $3 \times 3$ board has 4 purple cells, there are a total of $1681 \times 4 = 6724$ purple cells.