Expected number of pieces of a chessboard

Question

If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into?

Can anybody please provide how can I approach the problem? There are numerous cases and when I go through case consideration it becomes extremely complex.

Answer 1

2 = = 2 piece 3 = (3+1)/2 = 2 piece 4 = = 2 piece 5 = (5+1)/2 = 3 piece 6 = 3 piece 7 = (7+1)/2 = 4 piece 8 = = 4 piece

Series = (n+1)/2 pieces

Regards, Yuvaraj

Answer 2

I am posting this as an answer to make it more visible. This is currently a Brilliant.org problem, in which the specific case $ 2 \times 500 $ is asked for. While it is an interesting problem meriting discussion, that should wait until after 8:00 PM EST 5/19 when the problem set is closed.