The Computational Expressiveness of a Model Train Set: A Paperlet
scottaaronson.com
scottaaronson.com
It's my recollection that the very very very first computer hackers were members of the MIT model railroading club, because they understood electric circuits.
What are the advantages over using a computer?
As mentioned in the wikipedia article, the Steven Levy "Hackers" book has a great chapter on those early days.
See also the Jargon File entry:
https://en.wikipedia.org/wiki/Dr._Nim
It uses similar flippers to operate a marble-driven computer to play Nim.