This could be tough

I am told that this problem can be solved just by consideration of parity.
Let g(x) be a bijective function N->N and 'k' be any positive odd integer. Prove that there does not exist any function f(x):N->N such that
f (f(x)) = g(x) + k

Some old wine...

Well, take this old probability problem.
A mathematician carries two matchboxes(in order to complicate life!), both initially containing 'n' sticks each.Whenever a stick is needed, he takes it from one of the boxes(boxes are distinguishable)but never keeps count of matches.Once it so happens that he takes out one of the box and finds it empty, takes out the other and that is also empty.What is the probability this happening?