The paper "The mate-in-n problem of infinite chess is decidable" by D. Brumleve, J. D. Hamkins, and me was inspired by a question asked on Mathoverflow. The paper is available on the arxiv. Please see J. D. Hamkins' blog for the abstract or if you would like to post a comment, here's a short version of the abstract:
"Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. The main theorem of this article, confirming a conjecture of the second author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable."
I'm looking forward to hearing about other papers inspired by Mathoverflow.