Identify what is being proved based on solution to problem

Question

I can't identify what is being proved when I look at the solution manual for each problem and the solution doesn't refer to the problem statement directly, i.e. give the assumption and restate the conclusion.

How do I identify what's being proved or what's the purpose of the proof just by looking at the solution?

I think that if I can figure out how to go from the solution of the problem to the problem statement then I'll be able to do the inverse operation, i.e. proofs.

I find that the problems and the proofs are actually restatements of each other. The problem to prove is a summarized and really convoluted way to stating the proof.

Answer 1

It's just not possible, in general, to determine the problem statement given the solution, as printed in the solution manual.

For example, here is a solution:

The equation reduces, modulo $9$, to

$x^3\equiv5\pmod9$,

which has no solutions. Therefore, the original equation has no solutions.

There is simply no way to determine what the question was, since all of the information about the original equation, besides its residue modulo $9$, has been erased.

So, I'm not sure whether or not trying to deduce the problem given the solution is a valuable way to spend your time.

The source for the above solution is Gerry Myerson's community wiki answer to "Awfully sophisticated proof for simple facts." (Gerry's answer contains the problem, too!)