I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds, that seem like they'd be similarly easy to solve, but which turn out to be fiendishly hard like this one?
Especially ones that can be visualized easily geometrically like this one.
https://mathoverflow.net/questions/100265/not-especially-fam...
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.
https://en.wikipedia.org/wiki/Bellman%27s_lost_in_a_forest_p...
Warning: don't read these unless you want your next weekend to disappear:
http://wardsattic.com/joomla/Download/BellmanForestProblem.p...
https://www.maa.org/sites/default/files/pdf/upload_library/2...
P.S. Previously on HN:
https://news.ycombinator.com/item?id=18001449
P.P.S. I have now added PDF links to the Wikipedia article.
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?
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.
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.
Typically we view pendulums as “simple harmonic oscillators” - basically equivalent to a mass on a spring, only going left-to-right rather than up-to-down.
But this is only an approximation. The differential equation for a simple harmonic oscillator is
x’’ = ax
which has a simple solution - sum of a sine and a cosine, aka a wave. (Here x’’ = second derivative of x with respect to time, or acceleration)
But the equation of motion for a real pendulum (from a free body diagram) is much more complicated and has no exact solution:
x’’ = sin(x)
It’s a very strange function whose second derivative is equal to the sine of the original function, and can’t be expressed in ordinary means. Any high school physics student can derive this equation of motion from Newton’s laws, but you need the assumption sin(x) ~ x to make any progress finding a solution.
The space of all differentiable linear functions is a vector space, and the derivative is a linear operator on that space (so functions and derivatives follow most of the same rules as vectors and matrices). If D is the derivative operator and x is a function of t, the equation x''=x can be written as D^2 * x = x, or 0 = (D^2 - I)x = (D + I)(D - I)x. The dimension of the kernel of (D - I) is 1, and so is the dimension of the kernel of (D + I). So the space of solutions to the original problem has dimension 0, 1, or 2. But you can find two solutions, namely cos(i * t) and sin(i * t). Those solutions are linearly independent, so they span 2 dimensions. So all solutions are going to be of the form A * cos(i * t) + B * sin(i * t). The case for x''=ax is similar.
I haven't studied it well enough to say more than that about it.
It’s been a long time since I’ve studied ODEs and mathematical physics but this solution seems easier than I had guessed (in the sense that it’s accessible to advanced undergrads instead of specialists).
Generally mathematicians would consider the number that answers this problem to be “known” in the colloquial sense, even before this better form.
Maybe the 3n+1 problem? It's definitely easy to understand in a few seconds, and definitely fiendishly hard. I'm not sure if it seems easy to solve though :)
Another one that came to mind is about efficiently computing the permanent of a matrix[1]. Maybe understandable in seconds if you know how determinants work.
https://en.wikipedia.org/wiki/Goodstein%27s_theorem
But this is easily proved (using infinite ordinals, but it seems it could be proved just by coming up with a similar concept like the infinite ordinal arithmetic, basically providing an upper bound for each step of the algorithm and showing that there's an eventual maximum to these and then there's a monotonic decrease, and that the number of steps are always finite).
The culprit is that (most) education involves churning through problems and their often formulaic solutions, merely with increasing levels of difficult. You get used to simple problems having simple solutions by exposure. It would have been nice to understand as a child that there's so much simple stuff not figured out, in many areas.
But presenting open-ended problems in children textbooks sounds like a quick way to confuse and anger parents.
I too would love such a list. Can anyone point me if one does exist?
I can't formulate proper google search query, but I once read funny story where spies were planting a papers with a math problem like it written by a kid, very simple looking, but the answer had actually crazy numbers.
Describe the shape of the spiral path they trace.