black and white balls in the box A box contains $731$ black balls and $2000$ white balls. The following process is to be repeated as long as possible.
(1) arbitrarily select two balls from the box. If they are of the same color, throw them out and put a black ball into the box. (We have sufficient black balls for this).
(2) if they are of different colors, place the white ball back into the box and throw the black ball away.
What will happen at last? Will the process stop with a single black ball in the box or a single white ball in the box or with an empty box?
I am unable to decide how to start and in which direction? should we apply probability or what?

Doing this for small values of the number of balls quickly reveals the pattern. It turns out the total number of balls always decreases by one, and the number of white balls always keeps the same parity. So we necessarily pass through a state with a single ball, and as the number of white balls is always even, this last ball must be black.
