Question 6
On a $9 \times 9$ board some cells must be coloured red, at least one. For each colouring, let $P$ be the number of cells, coloured or not, that have an even number of red neighbouring cells. (Two cells are neighbours if they share a side.) Give a colouring of the board that has the smallest possible value of $P$ and prove that there cannot be a smaller value.
CLARIFICATION: 0 is even.
CLARIFICATION: 0 is even.