In mathematics, you can tell a lot about a function from its formula; in fact, you can tell everything. Well, maybe you can't see everything that a formula implies. It may require a real skill to be able to tease a formula into telling you some of the simple things that it implies.
Puzzle 1: Suppose we have a formula f(n) for a function whose domain and range is the natural numbers. To be more specific, for every natural number n from 0 up, you can evaluate the function f(n), and for every natural number m, there is always at least one natural number n so that f(n)=m.
Suppose further that the function satisfies the following rule:
f(f(n)) < f(n+1)
What is the value of f(129)?
I give up, show me the solution.