You have a bag that contains $b$ black balls and $w$ white balls. ($b+w>2$) You put your hand into the bag and pull out a pair of balls.
- If both balls are black, you throw away one and return one to the bag.
- If both balls are white, you throw them both away.
- If the balls are of different colours, you throw away the black ball and return the white ball to the bag.
You repeat this process $N$ times. At the end, you end up with exactly one ball in the bag.
For what all values of $b$ and $w$ is the maximum possible $N$ different from the minimum possible $N$?