Proving P = NP without mathematical statements / computer program
A Turing machine, it must be remembered, is a kind of flowchart. So is the structure of a computer program generally. So turning "a&nbsp;flowchart" into a formal answer to the problem should be fairly easy, if it actually did work. Indeed, if one started with a terribly formal answer to P versus NP, most computer scientists would try to find a formulation of it which did come as close as possible to a plain english description in order to get as strong an understanding of the solution as possible.

But there is a fundamental problem with the sort of question that you're asking. What does it mean for someone who would be able to solve P versus NP — and by showing that they are equal, no less — to not actually be either a computer scientist or a mathematician? Perhaps they are not employed professionally as a computer scientist or mathematician, but this is beside the point if they have the skill to solve what some (Scott Aaronson, for example) describe as the most important mathematical problem we've ever considered. If someone has the training (or has even self-taught) to successfully tackle the problem, and also to clearly communicate the solution to others by identifying the major sub-routines and their roles in solving e.g.&nbsp;SAT or HAMPATH, then whether they are employed or even have degrees is an irrelevant detail; they are nevertheless in that case a mathematician or a computer scientist. Better still if they can describe how their solutions overcome classic obstacles such as oracle results, such as oracles A for which PA &ne; NPA (or the opposite) by showing specifically what sorts of structure in the problem the algorithm takes advantage of, which would not be accessible in the oracle model. The problem, however, is that most people who dream of solving P versus NP as amateurs or outsiders seem to lack the communication skills to actually describe their work adequately, or (by virtue of not having read enough) they are unaware of results which would make their approach to solving the problem doomed from the start.

As with all dreams of glory these days, there is a basic problem with the fantasy of being the one to resolve P versus NP. The problem is that it's bound to be nearly impossible. Not actually impossible, mind you, or at least not necessarily impossible; just nearly so. As someone bright with ambition, it is possible  for one to lose sight of the fact that there are many other bright people: many of whom have also thought about the problem; and many of whom are brighter than oneself, even by a couple of orders of magnitude. And that there have been such bright people for as long as the problem has been around; and yet it remains unsolved. Yes, it's possible in principle that everyone is thinking about it the wrong way, and have been for decades. But is that really particularly likely? Nobody should expect themselves to be the one person who can spot the one sign-error that everyone else is making, because if everyone else is making that error then there must be something about the problem that will lead one to make the same mistake. Or — in the more likely event that the reason why the problem remains unsolved is not that people keep making simple mistakes or haven't yet thought of the one simple trick that dissolves the whole thing — what makes the problem fundamentally difficult is essentially an objective difficulty of the problem, and no clever dancing steps will allow one to simply waltz gracefully past all obstacles; that what is required is an approach which is not merely novel, but quite profound, identifying subtle structures that there was good reason for nobody to have seen before. The sort of structure which one is most likely to spot by thinking continuously about the problem for years.

If you want to be realistic about what it would take to solve the P versus NP problem, you might compare it to similarly famous breakthroughs in the past few decades, such as the proofs of the four-colour theorem, Fermat's Last Theorem, or the Poincaré conjecture. They might have simpler proofs someday, but the original proofs take you far into the wilderness to get you to the end (or in the case of the Four Colour theorem, the route is very long and repetitive). There's no particular reason to suspect that P versus NP will be different; so that if in the end it is resolved by an amateur, chances are extremely strong that it would be by someone with similar background knowledge and awareness of techniques of someone who is academically trained. Any realistic amateur who dreams of solving P versus NP would do well to keep that in mind.