I'd like to create a background of a gnomish teacher.  The character must be a gnome or raised in a gnomish culture in order to qualify for this background.  It is a combination of sage (usually of some highly eccentric area of knowledge) and entertainer (but a very specific kind of entertainer in as much as they use riddles, games,, tricks, puzzles, etc. as a means to teach).

Well, just give homebrewing it a try and have fun!

This is an entertaining idea. Proficiency with some specific gnome games, deception, and prestidigitation seem like a place to start.