Question 43
Let $n,k$ be positive integers with $k\le2n^2$. Lee and Sunny play a game on a $2n\times2n$ square grid. First, Lee writes a nonnegative real number at most $1$ in each cell, so that the total of all entries is $k$. Sunny then cuts the paper along grid lines into pieces, each consisting of whole cells and having total at most $1$. There are no other restrictions on the shape of a piece.
Let $M$ be the final number of pieces. Lee wants to maximize $M$, while Sunny wants to minimize it. Find $M$ when both play optimally.
Let $M$ be the final number of pieces. Lee wants to maximize $M$, while Sunny wants to minimize it. Find $M$ when both play optimally.