It's fascinating how a problem that looks at first glance like it belongs as a basic homework problem in a calculus textbook turns out to be so difficult.
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.
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.
One of my favorite classic examples is... a pendulum!
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.
I remember a take-home physics test in high school where there was a pendulum problem and I spent a full day trying to prove that x’’= ax is generally met by a trigonometric function. I eventually gave up, and was very surprised when I got full marks for saying to simply assume it (though the rest of the test suffered by my being stuck for so long). Besides the exponential formulation of sin and cos, I’m honestly not sure if that’s a unique answer to this day. Does anyone know?
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 agree with the main idea of your argument, but how can it be set in the "space of all differentiable linear functions" when trigonometric functions aren't linear? I think the argument would hold on the space of real analytic functions. Maybe that is more restrictive than necessary.
Ah yes that was a slip up. You're right, it should read the "space of all differentiable functions," or maybe "all complex differentiable functions" for extra precision.
Thanks - I wasn’t actually aware of this but I meant to write “elementary” rather than “exact” so I’m glad you brought this up. I had supposed there was probably some incantation of special functions that solved the equation :)
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).
There’s a lot of simple geometric problems like this that don’t have known closed form solutions. A lot of simple geometric time to intersection problems (comes up a lot in game physics) don’t have known explicit solutions for example. Finding them is interesting but once you have something you can use an iterative algorithm on to find a solution at arbitrary precision the pressure/incentive to solve them is reduced quite a lot.
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.
3n + 1 is interesting because it seems so tantalizingly easy and beautiful at first glance. Its like looking at a stream where you can clearly see the stony bed and it appears to be ankle deep but once you take a step you realize the lighting has fooled you and you fall head over heels into icy waters.
It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their termination at 0.
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).
Kepler's equation is a bit like this [1]. A body is moving in a known elliptical orbit, and you would like to know where it is at any given time - eg a quarter of the way through its orbit (so going from 'mean anomaly' to 'eccentric anomaly'). The starting point is a handful of simple equations, of motion and gravitation. But there is no analytical expression that answers the question - you have to solve it numerically, or approximate it with a sufficiently accurate Taylor series.
>at first glance like it belongs as a basic homework problem
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 have discovered a truly remarkable proof of this theorem which this margin is too small to contain."
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.
Put a person each at the corner of a square, each facing the person in the corner in the clockwise direction. Have each walk at the same time at the same speed toward the person they're facing, so they all keep adjusting their direction as the person in front moves.
Most Diophantine equations. Someone has already mentioned FLT above. Problems of this sort is almost one of the defining characteristics of number theory.
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.