This math.se comment by Daniel Wainfleet offers a problem that I found fun.
M. Wainfleet quotes the author (Rouse Ball or Coxeter):
This is a very useful hint!
Mouse over to view the solution:
Since !!p!! and !!q!! are both odd, their mean, !!\frac12(p+q)!!, is an integer. It cannot be prime, since !!p!! and !!q!! are consecutive primes, so !!\frac12(p+q) = ab!! for some !!a!! and !!b!! each greater than 1. Then !!p+q = 2ab!!.