Brute force or computer simulations are often brought to bear on mathematical problems. Among the motivation for using these approaches is they can obviate the need for careful thought. But careful thought often shows a simpler path to the answer. Occasionally, by argument from analogy to already solved problems, or invoking properties of the type of expression, for example, that the answer must possess, a solution can be found without seemingly solving the puzzle at all. Or, if an equation must be solved, it's one much simpler than first thought. Some people refer to these solutions as coming to them in what they call "Aha!" moments.
I will post a series of "Aha!" puzzles, starting with a rather easy one.
The Golden Ratio g has a number of definitions. The one that provides the best clue to our answer is to note that a rectangle of dimensions g x 1 has the property that if it is cut into a square and another rectangle, the second rectangle has the same proportions as the first.
Prove that g is irrational.
I know its usual symbol is Greek phi; but I don't know how to make one here.