Coin on Chessboard

Let's start by looking at only one square. If the coin bleeds into another square, then it touches both black and white, so we want $1-$ probability that it lands entirely in a square.

Since the coin has radius $\frac{1}{2}$ , we then know that if the center of the coin lands more than $\frac{1}{2}$ units from the border of the square, it will be entirely within the region.

This gives us a $1 \times 1$ square that the coin can land in, so the probability it lies in more than one square is $1-\frac{1^{2}}{2^{2}}=\frac{3}{4}$ which gives us our answer of $3+4=\boxed{7}$