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.