I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math problems (proofs, probably) that on the surface look equally challenging. But…
- One is simple to solve (or prove)
- One is complex to solve (or prove)
- And one is impossible
So if a mathematician can’t simply look at a problem and say, “I can solve that in a day, or a week, or a month, how can anyone else that is truly solving a problem? The very nature of problem solving is that we don’t know where the solutions lies and therefore we don’t know how long it will take to get there.
Any input or suggestions would be greatly appreciated.
Best Answer
This isn't exactly what you're asking for, but it should serve the same purpose very nicely.
Hilbert gave a talk in 1920 or so in which he discussed the difficulty of various problems.
He said that great progress had been made in analytic number theory in recent years, and he expected to live to see a proof of the Riemann Hypothesis.
Fermat's Last Theorem, he said, was harder; maybe the youngest members of his audience would live to see a proof.
But the problem of determining whether $2^{\sqrt2}$ is transcendental was so hard that not even the children of the youngest people in the audience would live to see a solution to that one.
With the benefit of hindsight, we can see that Hilbert had it exactly backwards.
$2^{\sqrt2}$ was settled in 1929 - Hilbert lived to see it.
Fermat, as we know, held out until the 1990s.
And the Riemann Hypothesis is still unsettled.
The point of the story is not to make fun of Hilbert. The point of the story is that if even Hilbert, the strongest mathematician of his era, could be so wrong in judging the relative difficulty of various mathematical problems, then it must be a really hard thing to do - which, I think, is the point you are trying to make.