>Professional mathematicians don't guess answers to theorems.

Believe it or not, proving theorems is not the goal of many mathematicians. I'm simplifying a bit, but read Freeman Dyson's essay on Birds and Frogs. Essentially "problem solvers" vs "theory builders". While problem solvers often do end up proving theorems, it is not their main goal. If they can "guess" a solution, they are done. It is publishable.

Go to the field of combinatorics, and you'll find it is full of guesswork.

>I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.

Sorry, but many mathematicians disagree with you. Solving a problem is finding a solution (provided you have a means to verify correctness). It doesn't matter if you merely guessed it.

My training was in commutative algebra as a pure mathematician. I did dabble in combinatorial commutative algebra and went so far as purchase a book on the topic by Sturmfels and Miller. I doubt very much that you can find a published paper in mathematics in which the author guessed the answer to something and did not prove anything. Every paper I read (commutative algebra) proved things or discussed something heuristically with the goal of finding a method of proving.

I have the book by Stanley on combinatorics. There are not results of the form: I guessed A is the answer and it's right. Let's move on. This is does not happen. When one notices something is a solution the mathematician always wonders why. A mathematician wonders what underlying structure there is. Never is one satisfied by a guess.

If someone found a counterexample to the Riemann Hypothesis the first question would be, why is this number a counterexample? What caused the obstruction? You could problem publish a paper that just said, A is a counterexample to the Riemann Hypothesis. But you could not publish a paper that said, I guessed A is a solution to B and it turns out I was right.

Even in commutative algebra, there is a fair amount of guesswork. If I tell you that 2+3=4 in an abelian group with {2,3,4}, then ask you what is 2+4, you will immediately guess 2+4=2. Its just what the millennials term "stupid obvious". Yeah you can do a lengthy proof on why 2+4=2. Proof: Its 2 because 4 must be the additive identity, and 4 is the additive identity because 2 & 3 are not, and they aren't because if 2 was then 2+3=3, and if 3 was then 2+3=2, but because I've told you 2+3=4, by closure 4 must be identity, which implies 2+4=2. That's the whole story. Literally no mathematician I know will sit down & write that lengthy proof I wrote. They'll just tell you 2+4=2, and if you pester them with "But why?" they'll say "Because" and excuse themselves :)

In an elementary group theory course this is the type of problem that would be given when groups are first introduced. It's a good homework/test question and just writing a*c = a as the answer would not suffice (depending on the level of the course and aim of the problem). The point at that level is for the student to learn how to justify their beliefs.

This is especially so if one were in a basic mathematical logic course. Of course, in an algebraic topology course where this group showed up it would be assumed that everyone knows how to find the answer and why. No justification would be needed.

The paper you cited supports what I've been saying. You can publish a paper that says, "Here is a counterexample." You can't publish a paper that says, "Here is a solution I guessed to be correct."

Any method of finding a counterexample is accepted. Guessing a solution is not.

EDIT: The paper linked to was published because it was a counterexample to a famous conjecture.

So you're saying that if they left out the "search on CDC" line the paper would not have been accepted.

No. I'm saying it would not have been published if it wasn't a counterexample.

What is the "answer" to a theorem? If by "answer" you mean the conclusion of the theorem, then mathematicians guess these all the time; such guesses are generally referred to as conjecture.

A conjecture is someone, usually a well known expert in the field, saying, "I think this is true but have not been able to prove it." It's not considered a theorem until someone proves it. A famous example is the Goldbach conjecture. No proof has been found but people have been searching for a proof for a long time but so far no one has proven it.