An open-and-shut case is one whose outcome is immediately clear. I've never heard of a put-and-shut anything."open-and-shut" is a newpaper's expression which I remember "...open-and-shut case, ..." is original sentence wich is my impression. " open-and-shut" seems to be, I understand, a kind of manner which help people understand something easily and fastly.
So I consider to use this expressing manner when I meet to need explaining a difficult thing, i.e I mention " put-and-shut"