Mathoverflow has their "long-open problems which anyone can understand". This includes things like the integer brick problem: Is there a brick where all its dimensions (width, height, breadth, face diagonals and main diagonal) are integers? And Singmaster's conjecture: How many times can a number (other than 1) appear in Pascal's triangle?
One of my favorites is Bellman's Lost in the Forest Problem. It is a 2D geometric problem that is easy to state, understand, visualize and draw. The escape path is unconstrained, so there must often be a series of rules and decisions to be made. Some doodling quickly reveals its subtlety.
It is also nicely phrased as a class of problems, because the forest's size and shape are known to the victim, but there are no constraints on what the shape might be. Some of the classes are solved, so you can chase down the spoiler solutions, but others are still open.
If an ideal observer looks at the end of an ideal infinite cylinder, which is not deformed by perspective, when the cylinder is pointing directly in his line of sight he will see just a circle, but if the cylinder deviates slightly what will he see? A cylinder with finite length in his field of view? Or a cylinder that goes infinitely out of his field of view?
This doesn't seem to belong in the category of "problems anyone can understand". What do you mean by looking at a cylinder "which is not deformed by perspective"?
The only way I can think of to interpret this is that you pick a plane and project the cylinder onto it. (That's how you get the circle). But that's easy to do. Failing that, this looks like a "puzzle" that's supposed to sound interesting without actually meaning anything.
"which is not deformed by perspective" = parallel projection, as opposed to a conic projection. The further from the closest cap on the cylinder its apparent size remaining the same as the cap. In a deformed projection the opposite cap in infinity would collapse to a point. And that would defeat the thought experiment.
What is the thought experiment? It's easy to know what the coordinates of the cylinder are. It's easy to project them however you want. Doing this doesn't tell you much. For the actual problem, we also have to ask:
- Is the rest of the world seen normally, or is it "not deformed by perspective" either? If there's no perspective for anything, most of what you see will just be blank white, the effect of the many different representations of different distant objects all overlapping one another in your vision.
- How are you "looking at" this cylinder? You're not using your eyes.
I don't remember the source, the cylinder just helped to visualize the problem, probably you can replace the cylinder with a line. The idea is that you can see its whole length.
https://mathoverflow.net/questions/100265/not-especially-fam...