This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with plain old algebra, which yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes:
The numbers $1-12$ are to be placed around a circle, as on a clock, but in any order. Show that there are three consecutive numbers in the arrangement with a sum of at least $19$.