Let's say we're trying to solve a jigsaw puzzle of $N$ pieces, but we're using "brute force". Our strategy is as follows: we pick a spot and then keep trying puzzle pieces until we hit the one that belongs on that spot. We somehow know, after we place a piece, whether it belongs on that spot or not. If for a moment we don't care about the orientation of those pieces (i.e. we know in advance what the right orientation of all the pieces is) or about side and corner pieces (i.e. they don't exist in our puzzle world), what is then the expected number of moves we need to make before we finish our puzzle?
Edit: what if we did care about orientation, would the new expected number of moves be just four times the old expectation value?
And what would happen if we changed our strategy to picking one piece and trying all the spots it could be in? Would the expectation values be the same?
What if we'd include the corner and side pieces in our problem? So lets say the puzzle has $b \times w = N$ pieces. Would the expectation value decrease?
It's alot of questions but I was curious and couldn't seem to figure it out on my own. Thanks in advance!