Since so many of you are baseball fanatics (and I assume know the game inside out), I thought I'd run a question by you.

A no-hitter still records as an official no hitter even if the pitcher walks some of the batters, right?   Does the no-hitter still stand if a run comes in by players that were walked onto base?   That has to be rare, but it must happen once in a while.  Does the no-hitter status get wiped out if a run comes in?  

A shut-out is a scoreless game.  So there can be hits, but no runs?  

Are both equal in terms of pitching difficulty in your estimation?   Do most shut-outs actually record very few hits, or are there any examples that you know where there were actually lots of hits, but the pitcher still managed a scoreless game?    

Thanks.   :)