Problem - Domino piling

You have an $M \times N$ board and dominoes that cover exactly $2 \times 1$ cells. Your task: find the maximum number of dominoes you can place on the board without overlaps. For example, a $2 \times 4$ board fits 4 dominoes.

This problem teaches you to recognize when complex problems have simple solutions. You might think you need to simulate placing dominoes, but there's a direct mathematical approach that runs instantly.

Before reading the solution, think about this: what relationship exists between the board's total area and the number of dominoes you can fit?