Wikipedia defines a “guesstimate” as “an estimate made without using adequate or complete information, or, more strongly, as an estimate arrived at by guesswork or conjecture.” Guesstimation problems are problems which involve finding guesstimates. Examples of such problems include:
$1.$ How many people in the world have access to the internet?
$2.$ What is the number of people who own a piece of clothing that is primarily of the color red?
$3.$ What is the number of cigarettes smoked in a day, in the world?
I was assigned in class to come up with a “guesstimate” problem within pure mathematics. Something which immediately came to my mind was finding the number of possible topologies for a finite set, since there is no general formula for determining the number of topologies on a finite set. But my professor deemed this to be a bit too deterministic to be a guesstimation problem, arguing that since the number of subsets of the power set is finite, this makes it amenable to a straightforward algorithmic checking of combinations of such subsets to see if a topology is formed.
So, my first question is:
Is it possible to formulate a guesstimation problem on the number of topologies on an infinite set, such that the number of topologies is known to be finite, yet is not known to be easily determinable?
I mean, it is known that there are uncountably many topologies that can be defined for an infinite set. But is it possible to impose conditions on the topological spaces so formed, (like being $T_2$, $T_3$, or $T_4$ spaces) so as to make their number finite, yet not so easily determinable (amenable to computation using a rather straightforward algorithm)?
Secondly:
What are some other guesstimation problems within pure mathematics?
Some other problems which came to my mind were from the field of graph theory and combinatorics, like computing values of Ramsey numbers, but almost all of them seem to be longstanding open problems. I’m afraid I might end up choosing a guesstimation problem for which I hardly have any knowledge of the “tools” for neither “guessing” nor “estimation”. Can someone please help me out with suggestions? I understand that this is a borderline “project suggestion” question, but since I was assigned this in class, I don’t know who else to ask.
"How many primes are between $M$ and $N$, where $M$ is some big number and $N$ is an even bigger number?"
Currently, it's impractical to compute the answer to this question in general. You can use as leverage some behaviors about the density of primes in an estimate to put bounds on the answer.
But "how many?" That's a guess. You can weigh how close to one bound or the other you go, with more or less confidence, but it's still just a guess.