This is a General Questions type of question, but since it is subjective, it is probably going to be relocated to IMHO anyway:

What are some of the most difficult cognitive/cerebral tasks ever achieved by a human?

Discovering or perfecting some particularly obscure and arcane field of theoretical mathematics?

Inventing a particularly complex form of nuclear-energy technology?

Proving or debunking a math theorem (i.e., something like the Reimann Hypothesis, although that one hasn’t been solved yet?)

The question is a little arbitrary but the proof by Andrew Wiles of Fermat’s Last Theorem seems to fit (it is doubtful that Fermat himself ever had a real one because the only known solution is 150 pages of cross-disciplinary math principles that weren’t fully developed in his time).

https://en.wikipedia.org/wiki/Wiles’s_proof_of_Fermat’s_Last_Theorem

I thought it was pretty awesome when the human beat the mega computer in a chess game.

It’s a virtual certainty. Fermat wrote his note about a proof in a book in 1637. But he never published the proof despite the fact that he lived another twenty-eight years. Fermat published his discoveries; there’s no reason he wouldn’t have published his proof if it had worked out.

The likeliest explanation is the obvious one. Fermat had a good idea one evening and jotted down a note about it. But he later worked on it and discovered the proof he had thought of didn’t hold up to rigorous examination. This isn’t too surprising; there have been several pseudo-proofs.

So Fermat dropped the matter and went on to work in other areas. He never would have guessed that after his death people would find his offhand note in his book and make it famous.

Are you talking about Deep Blue? If so, that was in 1996 but Kasparov lost the match even though he did win individual games. That was a long time ago in computer terms though. Computers today can beat anyone almost every time at chess.

Researchers have finally moved on to even harder games to crack like the ancient game of Go. They finally succeeded this year when Google designed AI defeated the top ranked player in the world.

Real people are still better at many things than the best AI but the list is getting shorter all the time.

I’ve just finished reading ‘Station X’ by Michael Smith about the codebreakers at Bletchley Park in WW2, its pretty much nothing but a series of incredible mental feats, and achieved under life or death pressure as well.

Einstein’s general relativity, perhaps? I think if he hadn’t done special relativity, someone else would have and within a few years. But general relativity might still not have been discovered even today, without Einstein. Although I guess the failure of the GPS system (which would drift something like 8 miles a day without taking general relativity into account) might have led to the equations at least. Einstein was not actually the first to publish those equations, incidentally. Poincare beat him by something like 5 days. But Poincare learned of the general theory from Einstein and they were in correspondence.

The classification of the minimal simple groups by Thompson and Veit took something like 200 pages and has not been significantly shortened. Of course, that was the work of two. The complete classification all the finite finite simple groups takes about 15,000 pages, but many many people were involved. Also not been significantly simplified.

Agree that maths is the boss here - with some proofs you must be talking about thought processes and ideas that are new to the human mind, and only a vanishingly small subset of our species can understand them.

A more relatable example might be blindfold chess - where masters play many games of chess simultaneously with no sight of the board or pieces, just mentally visualising the game. The wiki article makes this sound almost mystical, with ‘health concerns’ leading to its banning in Soviet Russia in the 1930s, and it stimulating research in psychology.

Playing 20 boards blindfold is obviously a mammoth display of mental strength, but I could see a Master whipping 20 club players in his mind’s eye as perhaps not all that amazing. Taking on other Masters, though, where you’re really going to be wrestling with the position, seems way out there.

I read somewhere that with some of the prizes offered for solving difficult math problems, the issue was that a mathematician could solve the problem, but encounter some difficulties claiming the prize, since there might be only a very few people in the entire world who would understand what he/she was trying to say, or could verify that he/she had truly solved it or not.

I agree with your post, but you meant David Hilbert, not Henri Poincare, both incredible geniuses in their own right.

I’ve always considered these handful of people who perform human echolocation to be a near supernatural feat.