Uninteresting questions with interesting answers [closed]

Question

What are best examples of questions in mathematics that are not interesting until one knows the answers, whose answers themselves are what is interesting?

The thing that prompts me to post this is just one example. I've seen others, but they escape me at the moment. Here it is:

A torus is embedded in just the usual way in $\mathbb R^3$. It has parallels of latitude and meridians of longitude. A curve that meets every parallel of latitude at the same angle, or, equivalently, meets every meridian of longitude at the same angle, is a loxodrome. Suppose that angle is so chosen, given the shape of the particular torus, that the loxodrome goes through all $360^\circ$ of longitude in just the length it takes to go through all $360^\circ$ of latitude, returning there to its starting point. (There must be some conventional terminology for describing these windings, but I don't know it.) The question is: What are the curvature and torsion at the various points along this curve? Doubtless some will consider this question interesting, but to me, and, I suspect, to many, the answer, because it is so unexpected, is where this starts to get interesting. The answer is that the curvature is constant --- the same at all points on the curve --- and the torsion is everywhere $0$. (And it's really easy to deduce from that the precise value of the curvature.) I believe this was discovered in the 1890s and is stronger than the celebrated theorem of Villarceau, published in 1848. Villarceau's theorem says that a plane bitangent to a torus intersects the torus in two circles. This proposition does not assume as a hypothesis, but rather has as a (trivial corollary of its) conclusion, that the curve lies in a plane.

Answer 1

Uninteresting question: find $$\int_0^1{x^4(1-x)^4\over1+x^2}\,dx$$ Interesting answer: $${22\over7}-\pi$$

Answer 2

Another classical geometry problem with a similar flavor to Michael's example:

Let $C$ be a smooth convex plane curve, let $L$ be a small line segment, let $P$ be the mid-point of $L$. Slide $L$ around the curve, keeping the endpoints on $C$, so $P$ traces out a curve $C'$ inside of $C$. What is the area between $C$ and $C'$? For any particular curve $C$, this seems (to me) to be a rather uninteresting calculus problem. What's interesting, of course, is that the answer is independent of $C$ and depends only on the length of $L$.

(Actually, one can mark any point $P$ on $L$, and then the area depends on the lengths of the two sub-segments of $L$.)

Answer 3

According to Gauss, Fermat's Last Theorem is an example.

Answer 4

Gerry Myerson's integral and Joe Silverman's geometry problem fall into the category of problems that seem uninteresting at first because they can be answered by a straightforward calculation that is not expected to yield any insight after the answer is obtained.

Another potential category consists of problems that seem hopelessly difficult but that turn out to be tractable. As an example, I propose the question, "What are all the finite simple groups?" Superficially, this might seem (almost) as hopeless, and therefore as uninteresting, as the question, "What are all the finite groups?" Only when you know that there is a nice answer that can actually be proved does the question reveal itself to be extremely interesting.

Answer 5

How many lines in $\mathbb{CP}^3$ meet four lines in general position?

Given the `linear' nature of the problem one may be tempted to guess $0,1$ or $\infty$. But the answer is actually 2.

The proofs are also pretty interesting:

(1) Degenerate into two pairs of intersecting lines; the two lines are the intersection of the two planes containing them, and the line passing through the intersection point. Then infer that the number is an intersection number (so topological), and therefore independent of the configuration. Making this last claim rigorous was one of Hilbert's problems.

(2) Use the fact that three general lines determine a quadric surface in $\mathbb{CP}^3$. The fourth line will intersect the quadric in two more points. Now draw the two lines as rulings on that quadric in those points. Here it is also easy to see that there cannot be more than 2 lines with this property.

Answer 6

Watson's integral. Seemingly uninteresting question: calculate $$W_S=\frac{1}{\pi^3}\int\limits_0^\pi\int\limits_0^\pi\int\limits_0^\pi \frac{dx\,dy\,dz}{3-\cos{x}-\cos{y}-\cos{z}},$$ produces truly amazing answer: $$W_S=\frac{\sqrt{6}}{96\pi^3}\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right)= \frac{\sqrt{3}-1}{96\pi^3}\left[\Gamma\left(\frac{1}{24}\right) \Gamma\left(\frac{11}{24}\right)\right]^2.$$ See http://link.springer.com/article/10.1007%2Fs10955-011-0273-0 (70+ Years of the Watson Integrals, by I. J. Zucker).

Answer 7

I hereby propose as one of innumerable possible answers to this question: Hilbert's 10th problem.

Doubtless it's an interesting problem, to those who are interested in that sort of thing; otherwise Hilbert would not have included it in his list. But to me, and again I suspect, to many, the answer is a lot more interesting than the question, partly, but not only, because it is surprising.

The problem is this: Is there an algorithm that given any polynomial in any finite number of variables with coefficients in $\mathbb Z$, correctly answers the question: is at least one tuple of integers a zero of this polynomial?

To understand the answer, let's establish some defintions:

• A set $S$ of members of $\mathbb Z^n$ is diophantine if there is some $m\in\mathbb Z^+$ and some polynomial function $f$ in $m+n$ variables $y_1,\ldots,y_m,x_1,\ldots,x_n$ with coefficients in $\mathbb Z$ such that $(x_1,\ldots,x_n)\in S$ if and only if $\exists y_1,\ldots,y_m\in\mathbb Z\ f(y_1,\ldots,y_m,x_1,\ldots,x_n)=0$.
• A set $S$ of members of $\mathbb Z^n$ is decidable if there is some algorithm that, given a member of $\mathbb Z^n$ correctly answers the question: Is this a member of $S$?
• A set $S$ of members of $\mathbb Z^n$ is semi-decidable if there is some algorithm that, given a member of $\mathbb Z^n$, runs forever if the input is not a member of $S$, and ultimately halts if it is a member of $S$.

Obviously a set is decidable if and only if both the set and its complement are semi-decidable. The existence of semidecidable sets that are not decidable was discovered in the 1930s by several people working independently (I think including Stephen Kleene, Alan Turing, Alonzo Church and maybe others?) and some of them are noteworthy sets, e.g. the set of all satisfiable formulas in first-order logic.

Obviously all diophantine sets are semi-decidable.

The result that laid Hilbert's 10th problem to rest is Matiyasevich's theorem:

All semi-decidable sets are diophantine.

An immediate corollary is that no algorithm of the kind sought by Hilbert can exist.

In 1970, Yuri Matiyasevich finished off the proof, which had been worked on over a couple of decades by Julia Robinson, Martin Davis, and Hillary Putnam.

Answer 8

One question which I think might be fitting is:

How many prime numbers are there?

After you learn about infinitude of primes you might think that there is nothing really much to say about the topic. But if we try to consider how many primes there are asymptotically, we reach a very interesting field of research, which I believe gave rise to the analytic number theory.