TITLE: A polyomino puzzle
QUESTION [12 upvotes]: Is there a polyomino such that it can be glued to an I-shaped pentomino and to a X-shaped pentomino to obtain the same polyomino?  
Or is there simple proof for non-existence of such polyomino? [Edit: See "I-shaped" and "X/(+)-shaped" pentominos below:]

REPLY [16 votes]: ${}{}{}{}{}{}{}{}{}{}{}{}{}{}$