It is hard to measure how "difficult" it is to solve mathematical problems. However, it occurs to me that Putnam exam questions are a somewhat standardized unit of mathematical difficulty. https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition Specifically, each year, the Putnam has 12 questions, each worth 10 points, to be solved in a total of 6 hours, i.e. 30 minutes per question. The questions are cooked up by experts with the intentional goal of making them so difficult that the median score on the exam (which is taken by good math undergraduate students Canada+USA-wide) is zero; but in such a way that the best 1% students in the country will score above about 42 points, and the best student in the country will score near 120. And this standard presumably is time-varying. Undergrads know different things now, than they knew in 1938. They as far as I know are not allowed to use electronic assistance on the Putnam, but perhaps one day that would/should be permitted, which if so would change things considerably. Pierre de Fermat presumably was considerably smarter than I, but he'd probably have no chance against me in a competitive math exam because he was so ignorant because math was just not very developed at that time. And some fans have been compiling online databases of all past Putnam problems. So consider 1 Putnam exam problem, as one "unit of difficulty." What, then, is the difficulty level of the hardest math problems that humanity has been able to solve? That 1 person has been able to solve? That you ever solved? Etc? It seems to me, if a problem requires solving about 5 sub-parts, each 1 Putnam difficulty, that's probably about the best I am capable of. It sort of grows on you exponentially. I mean, if I were given a Putnam problem, there is perhaps 50% chance I could solve it. (Depends how motivated I was, if I had a lot of motivation maybe 90%?) So anyhow, assuming the 50% figure, then if I had to solve 5 in a row to solve the uber-problem, presumably there would be about 1/32 chance I could. And then of course you'd have to figure out the 5 sub-parts. So in this sense, I'm not very smart. I mean, if an undergrad can solve something in 20 minutes, it wasn't all that hard. And if the best I can hope to do is equivalent to about 5 of that, not very impressive. Now we have the advantage of standing on the shoulders of giants, electronic assistants, and database searches to locate the previous works by the giants. Plus, there's the collaborative effect where, if the problem requires solving 10 Putnam-level subproblems, and some of them have already been solved by somebody else, then even though you could not have managed all 10 by yourself, you still might be able to do it. So then what is reachable? The "classification of finite simple groups" for example, is one of the hardest math problems that humanity has (allegedly) managed to solve, and certainly beyond the ability of any one human. How many Putnams of difficulty were involved? Hard for me to say. I would guess this is less than a KiloPutnam, but more than 50 Putnams? How soon do you think a computer AI will be able to compete on the putnam exam? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)