Question 16
On a $123\times123$ board, each cell is colored purple or blue subject to these conditions:
- Every purple cell not on the border has exactly 5 blue cells among its 8 neighbors.
- Every blue cell not on the border 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 $3\times3$ subboard within the $123\times123$ board. How many cells of each color can it contain?
b) Calculate the number of purple cells on the $123\times123$ board.
- Every purple cell not on the border has exactly 5 blue cells among its 8 neighbors.
- Every blue cell not on the border 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 $3\times3$ subboard within the $123\times123$ board. How many cells of each color can it contain?
b) Calculate the number of purple cells on the $123\times123$ board.