Not especially famous, long-open problems which anyone can understand Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand.  Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in my teaching, to motivate general education undergraduates, and early year majors, suggesting to them an idea of what research mathematicians do.
Meaning of "not too famous" Examples of problems that are too famous might be the Goldbach conjecture, the $3x+1$-problem, the twin-prime conjecture, or the chromatic number of the unit-distance graph on ${\Bbb R}^2$.  Roughly, if there exists a whole monograph already dedicated to the problem (or narrow circle of problems), no need to mention it again here.  I'm looking for problems that, with high probability, a mathematician working outside the particular area has never encountered.
Meaning of: anyone can understand The statement (in some appropriate, but reasonably terse formulation) shouldn't involve concepts beyond high school (American K-12) mathematics.  For example, if it weren't already too famous, I would say that the conjecture that "finite projective planes have prime power order" does have barely acceptable articulations.
Meaning of: long open The problem should occur in the literature or have a solid history as folklore.  So I do not mean to call here for the invention of new problems or to collect everybody's laundry list of private-research-impeding unproved elementary technical lemmas.  There should already exist at least of small community of mathematicians who will care if one of these problems gets solved.
I hope I have reduced subjectivity to a minimum, but I can't eliminate all fuzziness -- so if in doubt please don't hesitate to post!
To get started, here's a problem that I only learned of recently and that I've actually enjoyed describing to general education students.
https://en.wikipedia.org/wiki/Union-closed_sets_conjecture
Edit: I'm primarily interested in conjectures - yes-no questions, rather than classification problems, quests for algorithms, etc.
 A: There is a lot of number theory elementary conjectures, but one that is especially elementary is the so called Giuga Conjecture (or Agoh-Giuga Conjecture), from the 1950:
a positive integer $p>1$ is prime if and only if
$$\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod{p}$$
A: Is $e+\pi $ rational?
A: 
Is the sequence $(3/2)^n \mod 1$ dense in the unit interval? 

In the other direction, Mahler's 3/2 problem: 

Do all elements of this sequence with large enough index $n$ lie in the interval $(0,1/2)$?

It is known that $\beta^n$ is uniformly distributed modulo one for almost all $\beta>1$, but explicit examples of $\beta$ for which density holds are not known. This question seems to originate in work of Weyl and Koksma on uniform distribution.
Update: Since posting this answer I've attempted to find some references with which to flesh it out, with only modest success. The earlier paper I have identified which deals with this question directly is T. Vijayaraghavan's 1940 article On the fractional parts of the powers of a number, in which it is shown that the sequence $(3/2)^n \mod 1$ has infinitely many limit points. Mahler conjectured in 1968 that the answer to his question is negative. Jeffrey Lagarias' 1985 survey on the Collatz problem, The 3x + 1 Problem and Its Generalizations, includes a one-page overview of the literature on the distribution of this sequence. Flatto, Lagarias and Pollington subsequently proved that the diameter of the set of accumulation points is at least 1/3; Mahler's question would be answered in the negative if this is improved to "at least 1/2".
A: Some pages:
Open Problem Garden
The Open Problems Project  edited by Erik D. Demaine, Joseph S. B. Mitchell, Joseph O’Rourke
A: Can one prove the infinitude of the primes without employing any functions of super-polynomial growth?
(Of course I confess I have in mind Paris and Wilkie's more precise and sophisticated question concerning primes in the theory of bounded induction, but I think a high school student could think about looking for a positive answer without that background.)
A: The following problem is very well-known among algebraic geometers:
Does there exist a cubic 4-fold that is not rational?

It's probably not well-known outside of algebraic geometry, even though it can easily be explained in every elementary terms:
Does there exist a polynomial equation $F$ of degree three in five variables with the following property: Let $X \subset \mathbb C^5$ be the solution set of $F = 0$. Then there exists no chart $U \subset \mathbb C^4, \phi \colon U \to X$ such that $\phi$ is defined by rational functions (i.e., quotients of polynomials).
A: Waring's problem inequality
One of the oldest (Since 1770) and famous open problem in number theory is Waring's problem. It has been conjectured that if
$$
\left\{\left(\frac{3}{2}\right)^n\right\} \le 1 - \left(\frac{3}{4}\right)^n.
$$
(where $\{ \cdot \}$ denotes the fractional part) true then, the general solution of Waring's problem is
$$
g(n) = 2^n + \left\lfloor{\left(\frac{3}{2}\right)^n}\right\rfloor - 2.
$$
A: The easy-to-understand "equal sums of like powers" problem, which generalizes Pythagorean triples:
$$3^2+4^2 = 5^2$$
$$3^3+4^3+5^3 = 6^3$$

In general, does,
$$x_1^k+x_2^k+\dots +x_k^k=z^k$$
have a non-zero integer solution for all positive integer $k$?

So far, integer solutions are known for all $k\leq9$, except $k=6$.
(Unfortunately, the Eulernet search for $k=6$ has been stopped since the mid-2000s. With today's computers, and with a distributed search, it may be feasible to find it now.)
A: One I saw in a talk yesterday by Faustin Adiceam (I hope I have remembered this correctly):
Danzer's problem (in dimension 2): Is there a subset $S$ of $\mathbb{R}^2$ of finite density (the number of points at distance $\le r$ from the origin is $O(r^2)$) that hits every rectangle of unit area?
There is also a version where instead of positive density, a stronger condition is imposed: there is $\delta > 0$ such that any two points in $S$ are at least distance $\delta$ apart.
(Both versions are usually stated for convex sets, but the rectangle versions are equivalent, as any convex set of area $1$ is contained in a rectangle of area $2$ and contains a rectangle of area $1/2$.)
A: Given $n\in\mathbb N$, what is the smallest $k\in\mathbb N$ such that the harmonic number $H_k>n$?
It has been conjectured that for all $n$ the answer is $\lfloor\exp(n-\gamma)-1/2\rfloor$. See A002387.
A: https://en.wikipedia.org/wiki/Keller%27s_conjecture
From Wikipedia:
Keller's conjecture is the conjecture introduced by Ott-Heinrich Keller (1930) that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face.
Keller's original cube-tiling conjecture remains open in dimension 7.
Conjecture was shown to be true in dimensions at most 6 by Perron (1940a, 1940b). However, for higher dimensions it is false, as was shown in dimensions at least 10 by Lagarias and Shor (1992) and in dimensions at least 8 by Mackey (2002), using a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. Although this graph-theoretic version of the conjecture is now resolved for all dimensions,
Keller's original cube-tiling conjecture remains open in dimension 7.
The related Minkowski lattice cube-tiling conjecture states that, whenever a tiling of space by identical cubes has the additional property that the cube centers form a lattice, some cubes must meet face to face. It was proved by György Hajós in 1942.
Szabó (1993), Shor (2004), and Zong (2005) give surveys of work on Keller's conjecture and related problems.
A: Is there such $n\in\mathbb{N}$ that ${^n\pi}\in\mathbb{N}$? (see tetration)
A: Farideh Firoozbakht conjecture states that the sequence $P_n^{1/n}$ is strictly decreasing, where $P_n$ is the $n-$th prime number. This conjecture can also give simple proofs to many theorems related to Prime numbers.
It is not proved yet.
A: It is currently unknown if all triangles have a periodic billiard path.  (See, for example, https://en.wikipedia.org/wiki/Outer_billiards#Existence_of_periodic_orbits)
A: From "An Invitation to Mathematics":  

Are there any integer solutions to $x^3 + y^3 + z^3 = 33$?

I thought this might be a good candidate since that book was meant as a bridge from competitive Mathematics to research. There are a few other examples, but I am quoting only one here due to your requirement.
Edit: Such integers x, y and z have  been found.
A: I always enjoyed telling people about the Inscribed square problem :

Does every (Jordan) curve in the plane contain all four vertices of
some square?

Update: Here is a variation due to  Helge Tverberg:  Does every (polygonal) curve in the plane outside of the unit circle, contain all four vertices of some square with side length >0.1?  This version implies the original problem and lacks disadvantages pointed out by Tim Chow and Henry Cohn.  See Ville H. Pettersson, Helge A. Tverberg, and
Patric R.J. Östergård, "A Note on Toeplitz' Conjecture," Discrete Comput. Geom. 51 (2014), 722–738.
A: What is the least $S$ (if any) such that any subset of a plane of area $S$ contains $3$ vertices of a triangle of unit area?
A: *

*Is Hilbert's tenth problem for Diophantine equations in rational numbers decidable?

*Is Hilbert's tenth problem for Diophantine equations of power $3$ decidable?

*Is there a universal Diophantine equation of power $3$?

*Is there a universal Diophantine equation containing less than $9$ variables? If so, what is the minimal number of variables? What minimal power can be achieved for that number of variables?

*Is there a universal Diophantine equation that can be written using less than $100$ arithmetic operations (additions or multiplications)? If so, what is the minimal number of operations?

A: The list coloring conjecture: A list of colors is assigned to each edge of a finite graph $G$. A "list coloring" of $G$ is an edge-coloring such that (1) each edge is colored with a color from its list, and (2) edges that meet at a vertex have different colors. Suppose the graph $G$ admits a list coloring when the list $\{1,2,\dots,n\}$ is assigned to every edge; does it still admit a list coloring when an arbitrary list of $n$ colors is assigned to each edge?
A: The Kurepa problem: Show that for all primes $p>3$ we have that
$$
0!+1!+2!+\dots+(p-1)!
$$
is not divisible by $p$. Kurepa posed this problem in 1971. For an overview see the article by Ivic and Mijajlovic (https://arxiv.org/abs/math/0312202).
A: I like the Montesinos-Nakanishi 3-move conjecture from knot theory. A 3-move on a link is the replacement of two parallel strands by three half twists. The conjecture is that any link can be turned into the trivial link by a sequence of such moves. (If you think of this as a conjecture on diagrams, then you also need to allow Reidemeister moves.) According to an Encyclopedia of Mathematics article:

The conjecture has been proved for links up to 12 crossings, 4-bridge
links and five-braid links except one family represented by the square
of the centre of the 5-braid group. This link, which can be reduced by
3-moves to a 20-crossings link, is the smallest known link for which
the conjecture is open (as of 2001).

A: In number theory, the Odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd.
Stein, Selfridge, Graham, and others have posed the question of

whether the odd greedy algorithm terminates with a finite expansion for every $x/y$ with $y$ odd?

A: I really like this problem of Busemann and Petty (Problems on convex bodies, Math. Scand., 1956): 
Given a $0$-symmetric convex body $K$ and one of its support hyperplanes, construct the solid cone whose apex is a point at which the hyperplane touches the body and whose base is the central section of $K$ that is parallel to the hyperplane. Clearly the volume of this cone will be independent of the choice of contact point (in the case the support hyperplane intersects the body in more than one point). Now assume that no matter what support  hyperplane you choose, the volume of the cone is always the same. Is $K$ an ellipsoid?
As far as I know there are not even meaningful partial solutions to this problem (e.g., assume $K$ is three-dimensional and the boundary of $K$ is real analytic and strictly convex).
A: There are infinitely many primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$.
First explicitly asked by Gauss, now generally thought of as a corollary of Artin's primitive root conjecture.
A: The circulant Hadamard matrix conjecture, first stated in print by Ryser in 1963. It can be stated as follows. If $n>4$, then there does not exist a sequence $(a_1,a_2,\dots,a_n)$ of $\pm 1$'s satisfying
   $$ \sum_{i=1}^n a_i a_{i+k}=0,\ 1\leq k\leq n-1, $$
where the subscript $i+k$ is taken modulo $n$.
A: 
Problem: The partition function $p(n)$ is even (resp. odd) half of the time.

Of course you need to explain to a general audience what the partition function is, but that's not hard, my daughter in K1 got as an assignment to compute $p(n)$ for $n$ up to 4.
You also need to explain "half of the time", which means that the number of $n < x$ such that $p(n)$ is even, divided by $x$, has limit 1/2 when $x$ goes to infinity, so you need the notion of limit of a sequence, which is in K12, isn't it ?
The problem is certainly famous among specialists, but not too famous. I don't think there are books on it, for instance. It is old (formulated as a conjecture during the 50th), with an history going back to Ramanajunan. And I like it very much.
UPDATE (28/2/2015)
Here is a useful reference: 
Ken Ono, The parity of the partition function, Electronic Res. Ann. (1995)
A: Alexander's Conjecture, and by extension a lot of open problems about combinatorial subdivision, are as easy to understand as they are maddening. To quote Melikhov:

Alexander's 80-year old problem of whether any two triangulations of a [3-dimensional] polyhedron have a common iterated-stellar subdivision. They are known to be related by a sequence of stellar subdivisions and inverse operations (Alexander), and to have a common subdivision (Whitehead). However the notion of an arbitrary subdivision is an affine, and not a purely combinatorial notion. It would be great if one could show at least that for some family of subdivisions definable in purely combinatorial terms (e.g. replacing a simplex by a simplicially collapsible or constructible ball), common subdivisions exist...

Stellar subdivision (and arbitrary subdivisions) can be explained to a K-12 student with a picture. For a stellar subdivision, choose a face F, take its midpoint, and connect it to all vertices of tetrahedra of which F is a face. For arbitrary subdivision, invent some silly triangulation of a simplex, and just plug it inside. refining heighbouring simplexes as needed.
A: What is the least $V$ such that any convex body of unit volume can be fit into a tetrahedron of volume $V$? It is known that $V \ge 9/2$ and conjectured that $V = 9/2$.
A: More than ten years ago I posed the following problem in a couple of math-related mailing lists:

Let $G_n$ be the graph with vertex set $\{1, 2, \dots, 2n\}$ such that $\{i,j\}$ is an edge if and only if $i+j$ is a prime number. Is it true that $G_n$ is Hamiltonian for every $n \geq 2$?

It is a simple consequence of Bertrand's Postulate (there is always a prime between $k$ and $2k$) that $G_n$ is connected and has a perfect matching for every $n$.
The problem turned out to be an old one. I believe that some variation of it appears in Richard K. Guy's "Unsolved Problems in Number Theory" and according to this article, it was originally posed in the Journal of Recreational Mathematics in 1982.
Michael A. Jones and Leslie Cheteyan, "Two observation on unsolved problem #1046 on prime circles of $\{1, 2, . . . , 2m\}$", J. Recreational Mathematics Vol.35(1) (2006), 15--19.
A: Is there any odd perfect number?
A: Assuming that the definitions of a graph, its diameter and girth are something anyone can understand*, whether a graph with diameter $2$, girth $5$ and degree $57$ exists or not is a long standing famous open problem. See this or this.
There are several such (not especially famous) open problems regarding existence/uniqueness in the theory of bipartite Moore graphs (known as generalized polygons), i.e., graphs with diameter $d$ and girth $2d$, the prime power conjecture for finite projective planes that OP has mentioned being one of them. For example, is there a unique $4$-regular bipartite graph with diameter $6$, girth $12$, the so called generalized hexagon of order (3,3)?
*I have never had a problem with explaining this problem to people who have no math background beyond high school.
A: A conjecture arising from Waring problem: a number of solutions of the equation
$$x_1^3+x_2^3+x_3^3=y_1^3+y_2^3+y_3^3,\qquad|x_i|,|y_i|<P,$$
is $O(P^{3+\varepsilon})$. The best known estimate is $O(P^{7/2+\varepsilon})$ (Hua Loo Keng).
A: This requires some multivariable calculus, so maybe it is not strictly speaking to "everyone", but you could still use it when teaching undergraduates: the unique continuation for the $p$-Laplace equation.
Let $\Omega \in \mathbb R^3$ be an open domain. Suppose $u \in C^2(\Omega)$ and
$$
\nabla \cdot (|\nabla u|^{p-2}\nabla u) = 0 \ \ \text{in}\ \Omega, \quad 1 < p \neq 2,
$$
with $u\equiv 0$ in some open ball $B \Subset \Omega$. Then the question is to show that necessarily $u \equiv 0$ in all of $\Omega$. 
The real open problem is to show this for all weak solutions (which are known to be $C^{1,\alpha}$), but I think this is open also for $C^2$-functions; so posing this makes sense also without any knowledge above multivariable calculus.
A: At the risk of stretching my own rule, please allow that I could define "ring" for a high school senior.   Then I'd proffer this question I heard years ago from Melvin Henriksen:
Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity? 
A: Sendov's Conjecture

For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

A: Does the series
$\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$ converge?
(Taken from https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown where there are more such examples)
A: Does every nonseparating planar continuum have the fixed point property?
A: Are there infinitely many partition numbers divisible by $3$? See A000041.
A: (From Alon and Tarsi)
Let $A$ be a nonsingular $n$ by $n$ matrix over the finite field $\bf{F}_q$, $q>4$,
then there exists a vector $x$ in $\bf{F}_q^n$ such that both $x$ and
$Ax$ have no zero component.
A: I like Moser's worm problem: https://en.wikipedia.org/wiki/Moser%27s_worm_problem
I'm not sure how famous it is, but I believe less than it deserves.
In short: what is the area of the smallest planar region such that every curve of unit length can be placed into it?
A: Here is one which I found at this MO link:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq                         
     1} \left(\frac n7\right)\frac{1}{n^2}, $$
where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really
my favorite identity, but it has the interesting feature that it is a
conjecture! It is a rare example of a conjectured explicit identity
between real numbers that can be checked to arbitrary accuracy.
This identity has been verified to over 20,000 decimal places.
See J. M. Borwein and D. H. Bailey, Mathematics by Experiment:
Plausible Reasoning in the 21st Century, A K Peters, Natick, MA,
2004 (pages 90-91).

P.S. This problem was resolved before this post was placed in Section 5 of [D.H. Bailey, J.M. Borwein, D. Broadhurst and W. Zudilin,
Experimental mathematics and mathematical physics,
in "Gems in Experimental Mathematics", T. Amdeberhan, L.A. Medina, and V.H. Moll (eds.), Contemp. Math. 517 (2010), Amer. Math. Soc., 41–58]. In fact, the problem was solved even before its mentioning in the 2004 book; the details of the story can be found in the article.
A: The irrationality of Catalan's constant $G=1-1/3^2+1/5^2-1/7^2+\cdots$.
Remarks: Although Catalan's constant is certainly well-known, the irrationality is
the tip of the iceberg of a related conjecture of Milnor about the linear independence
over the rationals of volumes of certain hyperbolic 3-manifolds (which is a special
case of a conjecture of Ramakrishnan). The
irrationality of Catalan's constant would imply that the volume of the
unique hyperbolic structure on the Whitehead link complement is irrational.
To this date, it is not known that any hyperbolic 3-manifold has irrational
volume.
A: Here are a few others:

*

*Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$,
$$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$
Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!


*Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 0$. Then $x_n$ is unbounded.


*The largest integer that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers, is 462. It is known that there exists at most one such integer $n>462$, which must be greater than $2\cdot 10^{11}$. See J. Borwein and K.-K. S. Choi, On the
representations of $xy+yz+xz$, Experiment. Math. 9 (2000), 153-158; https://projecteuclid.org/journals/experimental-mathematics/volume-9/issue-1/On-the-representations-of-xyyzzx/em/1046889597.full.
A: The Kneser–Poulsen conjecture in dimension 3: An arrangement of (possibly overlapping) unit balls in space is tighter than a second arrangement of the same balls if, for all $i$ and $j$, the distance between the centers of ball $i$ and ball $j$ in the first arrangement is less than or equal to the distance between the centers of ball $i$ and ball $j$ in the second arrangement.  The conjecture is that a tighter arrangement always has equal or smaller total volume.  True in the plane, open in higher dimensions.
A: Here is another easy to state problem which is 140 years old but not very famous.
Consider the potential of finitely many positive charges:
$$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\quad a_j>0$$
How many equilibrium points can this potential have? Equilibrium points are solutions
of $\nabla u(x)=0$.
First conjecture: it is always finite.
Second conjecture: when finite, it is at most $(n-1)^2$. This estimate is stated by Maxwell
in his Treatease on Electricity and Magnetism, vol. I, section 113, as something known.
The editor
(J. J. Thomson) wrote a footnote that he "could not find any place where this result is proved".
Nobody could find this place to this time. This is even unknown in the simplest case
when all $a_j=1$ and $n=3$.
A: Let ${^n a}$ denote tetration: ${^0 a}=1, {^{n+1} a}=a^{({^n a})}$.

*

*It is unknown if  ${^5 e}$ is an integer.

*It is unknown if there is a non-integer rational $q$ and a positive integer $n$ such that ${^n q}$ is an integer.

*It is unknown if the positive root of the equation ${^4 x}=2$ is rational (ditto for all equations of the form ${^n x}=2$ with integer $n>3$)

*It is unknown if the positive root of the equation ${^3 x}=2$ is algebraic.

A: The continuum hypothesis.  Of course it's extremely famous, but everyone thinks it's resolved.  I was astonished to find out that some serious set theorists apparently consider it (I mean in the present, decades past Cohen's proof) to be an important open problem that people should be working on solving (for some meaning of "solve").
P. Koellner ( http://logic.harvard.edu/EFI_CH.pdf, Wayback Machine ) describes some current approaches.
A: Can a square with integer sides contain a point whose distance from each corner is an integer?
A: If $2^x$ and $3^x$ are integers for some positive real number $x$, does this imply that $x\in\mathbb{N}$?
A: It is not possible to find three corners of a square that form an equilateral triangle. (Left as an exercise for the reader.)
It is, however, possible to find four corners of a cube that form a regular tetrahedron:
                                          
We could ask: in which dimensions $n>1$ is it possible to do such a thing -  find a subset of the corners of the $n$-cube which form an $n$-simplex?
The answer: it is conjectured to be the case if and only if $n\cong 3\pmod{4}$, but remains open! The congruence condition is easily seen to be necessary, but existence of solutions for such $n$ has only been shown up through $n=663$ as of this writing.
(When you express the vertices of the $n$-cube as the set of length-$n$ $(0,1)$-tuples and do a little algebraic reshuffling, it becomes clear that this is equivalent to the Hadamard conjecture, a problem which was mentioned in an earlier comment but which I think becomes much more accessible under this framing.)
A: Is there a dense subset of a plane having only rational distances between its points?
A: 
Does there exist a point in the unit square whose distance to each of the four corners is rational?

This is sometimes called the rational distance problem, although that name often refers to a more general class of similar problems.  It's discussed by Richard Guy in  Unsolved Problems in Number Theory and in the following paper:
Guy, Richard K. "Tiling the square with rational triangles." Number theory and applications 265 (1989): 45-101.
It's also open whether there's a point outside the square whose distance to each of the four corners is rational, although it is known that no point on the edge of the square has this property.
A: Schinzel-Sierpinski Conjecture
Taken from this MathOverflow link.
Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:


*

*A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

A: Erdos's problem on the length of lemniscates (it is somewhat famous in certain narrow circles).
Let $P$ be a polynomial, and consider the set $E=\{ z:|P(z)|=1\}$ in the complex plane.

What is the maximum length of $E$ over all monic polynomials of degree $d$?

Erdos conjectured that an extremal $P$ is $P_0(z)=z^d+1$.
It is known that the asymptotic of maximal length is $2d+o(d).$
It is known that $P_0$ gives a local maximum. It is also known that for every
extremal polynomial, 
all critical points lie on $E$, so $E$ must be connected.
However the conjecture is not established even for $d=3$.
After Erdos's death, I offered a $200 prize for the first solution. (Erdos had offered the same, but I do not know whether one can collect his prize.)
A: A few decades ago Sherman Stein asked whether a trapezoid whose parallel sides are in the ratio $1:\sqrt2$ can be dissected into triangles, all of the same area. This remains open--it's a mystery which trapezoids admit such dissections.
A: I get the feeling that you will enjoy reading about the Simonyi and Chvatal conjectures described here by some guy called Gil Kalai. Anyone know who that is? ;)
A: N. M. Katz: "Simple Things we don't know": https://web.math.princeton.edu/~nmk/pisa16.pdf
A: Proving the Inequality of the Means by fitting boxes into a cube. From Berlekamp, Conway and Guy's Winning Ways for Your Mathematical Plays, Academic Press, New York 1983. See the discussion of this problem on Dror Bar-Natan's webpage for details, pictures, etc. 
Question: Is it possible to pack $n^n$ rectangular n-dimensional boxes whose sides are $a_1, a_2,\ldots, a_n$ inside one big n-dimensional cube whose side is $a_1+a_2+\cdots+a_n$? 
A: I think nobody pointed this problem, if it is repeated, please say me to delete it. This problem killed me for three weeks, when I was a young student in high school. So, I want to recall it again.
Problem: Find all right triangles with rational sides, where the area of these triangles are integer?
I think it is still open problem and if somebody can solve it, I will give 100$ as a small award.
After I searched, I found these two interesting sources. I hope it will be helpful. 
1) N.Koblitz, Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
2) Washington, Lawrence C.,  Elliptic Curves : Number Theory and Cryptography, CRC Press Series On Discrete Mathematics and Its Applications
A: The Cerny conjecture says that if X is a collection of mappings on an n element set such that some iterated composition (repetitions allowed) of elements of X is a constant map then there is a composition of at most $(n-1)^2$ mappings from X which is a constant mapping. This comes from automata theory. See https://en.wikipedia.org/wiki/Synchronizing_word.
A: Ramanujan's conjecture [*] If $2^x$ and $3^x$ are both rational (hereafter assumed) integers for some non-zero $x$ then $x$ is an integer.
[*] I think that is the accepted name for this problem. He certainly proved the weaker corresponding result with $2^x$, $3^x$, and $5^x$ all assumed to be integers.
Unlike some of the other fascinating conjectures already listed here, this one seems "obviously" true. Yet I gather little progress has been made on it. It must be hard to find a foothold, so to speak, or know where to start.
Another easily understood example is the Erdős-Straus Conjecture which
asserts that for every integer $n > 1$, there is at least one set of positive integers $x, y, z$ with $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{n}$. The result is trivially true if negative integers are also allowed.
In this case, by contrast, it's easy(ish) to "almost" prove it, and with patience and ingenuity one can proceed (apparently) ever closer to a solution. But a few annoying special cases always seem to slip through the net!
One more example - I think a high school kid would have little difficulty understanding the abc conjecture, or following the simple proof of the corresponding result for polynomials Mason-Stothers theorem.
A: This is basically copied from my answer on this question, which I've now updated some.
Let's let $\|n\|$ denote the smallest number of 1's needed to write n using an arbitrary combination of addition and multiplication.  For instance, ||11||=8, because $11=(1+1)(1+1+1+1+1)+1$, and there's no shorter way.  This is sequence A005245.
Then we can ask: For n>0, is $\|2^n\|=2n$?
Since it is known that for m>0, $\|3^m\|=3m$, we can ask more generally: For n, m not both zero, is $\|2^n 3^m\|=2n+3m$?
Attempting to throw in powers of 5 will not work; ||5||=5, but $\|5^6\|=29<30$.  (Possibly it could hold that $\|a^n\|=n\|a\|$ for some yet higher choices of a, but I don't see any reason why those should be any easier.)
Jānis Iraids has checked by computer that this is true for $2^n 3^m\le 10^{12}$ (in particular, for $2^n$ with n≤39), and Joshua Zelinsky and I have shown that so long as $n\le 21$, it is true for all m.  (Fixed powers of 2 and arbitrary powers of 3 are much easier than arbitrary powers of 2!)  In fact, using an algorithmic version of the method in the linked preprint, I have computed that so long as $n\le 41$, it is true for all $m$, though I'm afraid it will be some time before I get to writing that up...
That seems to be the best known.
A: 
Are there eight points on the plane, no three on a line, no four on a circle, with integer pairwise distances?

The analogous question for seven points was posed by Paul Erdős and answered positively by Kreisel, Kurz 2008, who have then asked this question.
In general, problems by Paul Erdős are worth to check if you want to find problems you are asking for here.
Tobias Kreisel, Sascha Kurz, There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle, Discrete & Computational Geometry 39/4 (2008), 786-790. (Wayback Machine)
A: The Kurepa conjecture : For every odd prime $p$, one has
$$
0!+1!+\cdots+(p-1)!\not\equiv0\pmod p
$$
A proof was claimed and published in 2004 but the claim was withdrawn in 2011.  See also my comment on the accepted answer to MO24265.
A: The Graph Reconstruction Conjecture:


Let $G, H$ be finite, simple, loopless graphs such that $|V(G)|$ and $|V(H)|$ are at least $4$. If there is a bijection $\varphi:V(G)\to V(H)$ such that for all $v\in V(G)$ the graphs $G\setminus \{v\}$ and $H\setminus \{\varphi(v)\}$ are isomorphic, then $G\cong H$.


A: Here's another Birch Swinnerton-Dyer related problem. Sylvester conjectured that every prime
that is 4,7 or 8 mod 9 is a sum of two rational cubes. Elkies (unpublished?) settled the first
two cases. As far as I know, the third is still open.
A: For integers $a_1, a_2, \ldots$, let $[a_1,a_2,\ldots]$ denote a continued fraction expansion $a_1 + \frac{1}{a_2 + \frac{1}{\ddots}}$.
Zaremba conjectured in 1972 that there must exist some constant $A > 1$ such that given any integer $d > 1$, you can always find some coprime $b$ such that if we write out the continued fraction expansion $b/d = [a_1, a_2, \ldots, a_k]$, all of the coefficients $a_i \leq A$.
Everyone seems to believe that $A = 5$, but we still don't know if such a constant exists. The current best known result (of Bourgain and Kontorovich) is that almost all integers $d$ permit a $b$ such that $b/d$ has the desired property (in their paper, they took $A = 50$, but I think that has been improved slightly since).
A: Does the decimal expansion of $2^n$ contain the digit $7$ for all sufficiently large $n$?
A: The Littlewood conjecture:


For any $\alpha, \beta \in \mathbb{R}$ we have $$\lim\textrm{inf}_{n\to\infty} (n\cdot||n\alpha||\cdot||n\beta||) = 0$$


where $||\cdot||$ denotes the distance to the nearest integer.
A: An open problem I find surprising, the PAC (Perimeter to Area Conjecture) due to Keleti (1998):

The perimeter to area ratio of the union of finitely many unit squares in the plane does not exceed 4.

See for example Bounded - Yes, but 4? and references therein.
A: One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the length of the main diagonal are all integers?
update 2012-07-12 Since the question has returned to the front page, I'm taking the liberty to add some links that I found after Scott Carnahan's comments. (Scott deserves the credit, really, but I thought the links belonged in the answer rather than in the comments.)

*

*On perfect cuboids, by Ronald van Luijk, master thesis, 2000.


*The surface parametrizing cuboids, by Michael Stoll and Damiano Testa, arXiv.org:1009.0388.
A: The complexity of matrix multiplication (i.e. the asymptotic number of steps required to multiply two n-by-n matrices).
This is an important problem in CS theory, but is non-famous enough in other fields that a mathematician (Andrew Stothers) made a significant advance in it in 2010 (beating a 20-year-old bound of Coppersmith and Winograd), and wrote up the result on page 71 of his PhD thesis without bothering to state it as a theorem or otherwise call attention to it.  Word of it only got around a year or so later, when a computer scientist (Virginia Vassilevska Williams) independently made a further improvement.
The obvious multiplication algorithm takes $O(n^3)$ steps, and a well-known Karatsuba-like rearrangement gets the exponent $\omega$ down to about 2.8.  There is a simple proof that the smallest possible $\omega$ is $\ge 2$.  Coppersmith and Winograd got an exponent of 2.376 and the more recent results have it at 2.373.  Apparently nobody has even shown that the minimum is not equal to 2: there are some who believe there's an algorithm faster than $O(n^{2+\epsilon})$ for any $\epsilon>0$ but not an $O(n^2)$ algorithm, but this is not known.
More info is in this blog post of Scott Aaronson: https://scottaaronson.blog/?p=839
A: Is there an upper bound of quotients in the continued fraction representation of $\sqrt[3]{2}=[ 1; 3, 1, 5, 1, 1, \dots]$?
A: Can a disk be dissected into two or more congruent pieces, with its centre lying within one of the pieces?
A: Bonnessen—Fenchel conjecture: Which convex body of constant width has the least volume? Is it Meissner's tetrahedron?
A: The following is a conjecture of Wlodzimierz Kuperberg:

Every convex planar set of area 1 is contained in a quadrilateral of area $1+\frac{4}{5}\tan\frac{\pi}{5}\sin\frac{\pi}{5}$.

In other words, such a set is contained in a quadrilateral of area less that $\sqrt{2}$, and the minimum is obtained for the minimum area quadrilateral containing a regular pentagon.
The conjecture involved only elementary plane geometry, and can be found in:
W. Kuperberg, On minimum area quadrilaterals and triangles circumscribed about convex plane regions, Elem. Math. 38 (1983), no. 3, 57–61, MR0703939 (85a:52009) 
It is presented as a challenge to the MO community here:
Small quadrilaterals containing a given convex region

It is easy to prove that
(*) Every convex planar set of area 1 is contained in a quadrilateral of area 2.
It is also easy to see that statement (*) remains true if the constant 2 is replaced with a somewhat smaller one. Contest: Find such a constant, the smaller the better.
Update:
Reaching $\sqrt{2}$ and even a strictly smaller value was proved by Chakerian (1973) and Kuperberg (1983) and the research challenge offered is to improve it even further, and perhaps even to verify the conjecture that the minimum is attained by a regular pentagon. But any nice arguments for bounds below 2 are welcome. 
A: 3D Version Of Blaschke-Lebesgue(1914) Theorem
The planar, compact.convex set of constant width, say 1, of minimal area is the Reuleaux triangle: Blaschk-Lebesgue(1914). The 3D set of constant width and minimal volume is unknown.
A: Is it true that any word of length $n$ contains less than $n$ squares? 
(A square is a factor of the form $uu$ for a non-empty word $u$.)
A: The moving sofa problem: What rigid two-dimensional shape has the largest area $A$ that can be maneuvered through an L-shaped planar region with legs of unit width?
So far the best results are $2.219531669\lt A\lt 2.37$.
A: Can we cover a unit square with $\dfrac1k \times \dfrac1{k+1}$ rectangles, where $k \in \mathbb{N}$?
(Note that the areas sum to $1$ since $\displaystyle \sum_{k \in \mathbb{N}}\dfrac1{k(k+1)} = 1$)
Here is an MO thread discussing some of the progress on this problem.
A: Here is a nice question due to John Conway. In a magical 4x4 square, show that the XOR composition of the four numbers, written in base 2, in every row and in every column is zero. This applies to a square in which the numbers 0 to 15 are used (rather than 1 to 16). 
For instance, a typical row might be 0 15 14 1, which in binary is 0000 1111 1110 0001, and in each of the four positions there happen to be two entries 0 and two entries 1, so the binary sum is zero.
Of course there are only finitely many possible magic 4x4 squares, and you can give proof by "complete inspection" (aka brute force). In fact, that has been done, so the result is true. But neither he nor I know a conceptual proof. Should be easy to understand about a classical problem -- and yet seems not obvious. Try it!
(Incidentally, the binary sum along the diagonals need not always be zero; that's not part of the question.)
A: This is the second time I've seen this question on MathOverflow and this will be the second time I've posted this answer.
Singmaster's conjecture says there is a finite upper bound on the number of times a number (other than the $1$s on the edge) can appear in Pascal's triangle.  The upper bound may be as low as $8$.  If so, then no number (besides those $1$s) appears more than eight times in Pascal's triangle.  Only one number is known to appear that many times:
$$
\binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6}
$$
It has been proved that infinitely many numbers appear twice; similarly three times, four times, and six times.  It is unknown whether any number appears five times or seven times.
Singmaster states that Erdős said the conjecture is probably true but probably difficult to prove.
A: A meta-answer: I recommend Guy's Unsolved Problems in Number Theory and perhaps some of his others (Unsolved Problems in Geometry, Unsolved Problems in Combinatorial Games), which have many unsolved problems (both well-known and obscure), grouped into categories. Many of these are of attackable difficulty.
A: The Happy Ending Problem

*

*Says that any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral. More generally, Erdös and Szekeres proved that for any positive integer $N$, there is a minimal integer $f(N)$ such that any set of $f(N)$ points in the plane in general position has a subset of $N$ points that form the vertices of a convex polygon, and it is known that $f(N)$ is at least $1+2^{N-2}$.

An open question is: does $f(N)=1+2^{N-2}$ hold?. Taken from this MO link.
A: In an oriented graph, is there always a vertex from which there are at least as many vertices that one can access by moving along exactly two edges, than there are vertices that one can access by moving along one edge?
This is known as Seymour's second neighborhood conjecture, and might be on the verge to being too famous (but it seems few of my colleagues know it).
A: Is the density of $1$s in the Kolakoski sequence $122112122122112112212112\dots$ (Wikipedia, OEIS) equal to $1/2$? Also, does every consecutive block, which occurs at all in the Kolakoski sequence, occur infinitely often?
A: Imre Ruzsa conjectured in 1971 (Mat. Lapok 22, in Hungarian) that a congruence-preserving mapping $f : \mathbb{N} \to \mathbb{Z}$ is a polynomial as soon as the power series $A(t) := \sum_{n \in \mathbb{N}} f(n)t^n \in \mathbb{Z}[[t]]$ has radius of convergence $> 1/e$. (Congruence-preserving simply means $n-m \mid f(n)-f(m)$.)
This is still an open problem, although A. Perelli and U. Zannier have shown that the power series $A(t)$ must be $D$-finite ("On recurrent mod $p$ sequences," J. reine angew. Mat. 348, 1984, DOI: 10.1515/crll.1984.348.135, eudml). The best result on Ruzsa's problem is due to U. Zannier ("On periodic mod $p$ sequences and G-functions," Manuscripta math. 90, 1996, eudml, DOI: 10.1007/BF02568314).
A: Are there infinitely many $n\ge1$ such that
$$ \gcd(2^n-1,3^n-1)=1 ? $$
Ailon and Rudnick conjectured that the answer is affirmative around 2000. What I like about this problem is that it could appear in Euclid, yet wasn't asked until fairly recently. There are obvious generalizations, and the analogue with $\mathbb Z$ replaced by $\mathbb C[T]$, is proven in the Ailon-Rudnick paper. There are also some (deep) results of Bugeaud, Corvaja, and Zannier giving the following related results:


*

*There is a constant $C>0$ and infinitely many $n\ge1$ such that 
$$ \gcd(2^n-1,3^n-1) \ge \exp(C n/\log\log n) . $$

*For every $\epsilon>0$ there is a $C_\epsilon$ such that
$$ \gcd(2^n-1,3^n-1) \le C_\epsilon\exp(\epsilon n) 
   \quad\text{for all $n\ge1$.} $$

A: Ron Graham [1,2] asked if there are infinitely many positive integers $n$ such that the central binomial coefficient $\binom{2n}{n}$ is coprime to $105 = 3 \times 5 \times 7$, and offered a prize of $1.000 for a proof/disproof. 
Accordingly to some heuristics [3, §4], there should be infinitely many such $n$, but if instead of $105$ a product of four primes is taken, than only finitely many such $n$ are expected.
Furthermore, it has been proved [4] that $\binom{2n}{n}$ if coprime to $pq$ for infinitely many $n$, where $p$ and $q$ are two fixed odd primes.
[1] D. Berend and J. E. Harmse, On some arithmetic properties of middle binomial coefficients, Acta Arith. 84 (1998), 31–41.
[2] OEIS, https://oeis.org/A030979
[3] C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.
[4] P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
A: The Feit-Thompson conjecture is not too famous and it is still open: there are no prime numbers $p\neq q$ such that $$\frac{p^q-1}{p-1}\text{ divides }\frac{q^p-1}{q-1}.$$
A: The lonely runner conjecture.  As Wikipedia puts it:

Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely if at distance of at least $1/(k + 1)$ from each other runner. The lonely runner conjecture states that every runner gets lonely at some time.

A: How many trees are there?
Let $T(n)$ be the number of trees on $n$ vertices up to graph isomorphism. There is no known closed formula for $T(n)$.
In 1947 Richard Otter proved[Source] the asymptotic result $$T(n) \sim A \cdot B^n \cdot n^{-\frac{5}{2}}$$ where $A \approx 0.535$ & $B \approx 2.996$.
By way of contrast, let $L(n)$ be the number of labelled trees, i.e. trees formed from vertices labelled $1,...,n$ where isomorphism additionally preserves the label. In 1889, Arthur Cayley showed[Source] that $$L(n)=n^{n-2}$$
A: Here is another problem on equilibrium points of potentials: suppose that we have infinitely many point masses in $R^3$ (the points do not accumulate). Must there exist a point where the
gravitational force created by these masses is zero?
If the masses $m_k>0$ are placed at $x_k\to\infty$ then the force is
$$\sum_{k=1}^\infty m_k\frac{x-x_k}{|x-x_k|^3},\quad \mbox{where}\quad\sum_{k=1}^\infty m_k|x_k|^{-2}<\infty.$$
Does every such function have a zero?
This is a version of the problem proposed by Lee Rubel in 1980-th. For some partial results
see  https://www.math.purdue.edu/~eremenko/dvi/equil.pdf
This question can be easily modified for any dimension $n\geq2$, using Newtonian ($n\geq 3$) or logarithmic potential ($n=2$). The question is substantially easier in dimension $2$,
but even for $n=2$ it is not solved in full generality.
A: A list of fundamental geometry problems with simple intuitive statements involving curves and surfaces in Euclidean space is maintained at:
Open Problems in Geometry of Curves and Surfaces
Five of these problems are listed below in no particular order (see the paper for references, background, and more precise statements):
Durer's Unfolding Problem: Can every convex polyhedron be cut along a collection of its edges and developed into the plane in a single non-overlapping piece?
Alexandrov's Conjecture on Intrinsic Diameter: Of all convex surfaces with a fixed intrinsic diameter (the distance between the farthest two points as measured within the surface), the one with the greatest area is the doubled disk (the degenerate convex surface obtained by gluing a pair of disks along their boundaries).
Zalgallers's Problems on Width and Inradius: What are the shorted closed curves in Euclidean 3-space with a given width or inradius? (The width is the smallest distance between all pairs of parallel planes which contain the curve in between them, and the inradius is the radius of the largest ball which is contained in the convex hull of the curve and is disjoint from the curve; see the post by Joseph O'Rourke).
Euler-Maxwell Rigidity Problem: Can one continuously deform a smooth closed surface in Euclidean 3-space without changing the distances between its points, as measured within the surface? 
Converse of the Archimedes Hatbox Theorem: Is the sphere the only convex surface such that whenever it is cut by a pair of parallel planes separated by a fixed distance, the area of the portion of the surface trapped between the planes is always the same? See this earlier post.
A: The Casas-Alvero conjecture: let the characteristic of the field $k$ be $0$. If a monic polynomial $f\in k[X]$ of degree $n$ has a common root with each of its derivatives $f',\ldots,f^{(n-1)}$, then $f(X)=(X-a)^n$ for some $a\in k$.
A: Gourevitch's conjecture1:
$$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$
1Jesús Guillera: About a New Kind of Ramanujan-Type Series; Experimental Mathematics (2003), Volume: 12, Issue: 4, page 507-510; DOI: 10.1080/10586458.2003.10504518,
eudml
A: Is there a positive integer which is both triangular and factorial except these obvious examples: $1, 6, 120$? (Tomaszewski conjecture, http://oeis.org/A000217)
A: Is there a rectangle that can be cut into $3$ congruent connected non-rectangular parts?
A: This problem is open (to my knowledge), surely accessible to anybody, and not too famous. As for the "Long open" requirement, there is at least a sizable group of people interested in it, as evidenced by this MO post:
Factoring 0-1 polynomials 
The problem is:
Is it true that if a polynomial $p(x)$ whose coefficients are in $\{0,1\}$ factors as $p(x)=p_1(x)p_2(x)$, where $p_1$ and $p_2$ are monic polynomials with non-negative real coefficients, then in fact also $p_1$ and $p_2$ have coefficients in $\{0,1\}$? 
A: Easy-to-Explain but Hard-to-Solve Problems About Convex Polytopes
slides by Jes´us De Loera contains 7 open problems (Hirsch conjecture is also there so it is out-of-date). 
A: I think you could give an accessible K-12 formulation of the definition of a group (as a group of permutations, for instance) and of an integral group ring. The Zero Divisor Conjecture (Kaplansky, 1940) then states, in one version, that if $G$ is a torsion-free group then the group ring $\mathbb{Z}[G]$ has no zero divisors besides the number $0$. 
A: Do there exist five positive integers such that the product of any two of them increased by 1 is a perfect square? 
The same question for seven distinct nonzero rationals.
Diophantine m-tuples pages
A: From Rick Kenyon's open problem list:
What are the minimal number of squares needed to tile an $a \times b$ rectangle?
Kenyon showed the correct order is $\log a$ assuming $a/b$ is bounded with $b \leq a$. However, there is plenty of room for improvement in the constant factor, and an exact formula seems far, far away.
A: Enumeration of meanders. (See also meander).
Problem is to find some formula for the number of meanders or at least some good asymptotic.
As far as I understand the attention to it has been attracted by V.I. Arnold.
The problem is so "everyone can understand" that there is an article by him in
the math. journal for shool-children "Quant" (sorry it is in Russian. I remember it from my school years):  djvu file from the site.
There are plenty papers in arXiv on the problem.
E.g. https://arxiv.org/abs/cond-mat/0003008
Exact Meander Asymptotics: a Numerical Check
Philippe Di Francesco, Emmanuel Guitter (SPHT-Saclay), Jesper Lykke Jacobsen (LPTMS-Orsay)
As far as I understand from the nice book (or) by S. Lando and A. Zvonkin
the problem is still open.
A: The following conjecture by Carsten Thomassen:

If $G$ is a 3-connected graph, every
  longest cycle in $G$ has a chord.

Thomassen has proven the conjecture true for 3-connected cubic graphs.
A: What is the largest possible volume of the convex hull of a space curve having unit length?
A: Let $R(x)=P(x)/Q(x)$ where $P(x)$ and $Q(x)$ are polynomials with integer coefficients and $Q(0)\neq 0$. Is there an algorithm that given $P(x)$ and $Q(x)$ as an input always halts and decides if the Taylor series of $R(x)$ at $x=0$ has a coefficient $0$?
A: The Polya--Szego conjecture for polygonal drums:  among the polygonal drums with $n$ sides and given area, the regular one has the slowest vibration (and therefore the lowest tone). 
As far as I know, this remains open for $n\geq 5$. 
A: The Alon-Tarsi Conjecture:
A latin square of order $n$ is a filling of an $n\times n$ matrix with the numbers $1, 2,\ldots,n$ such that each row or column gives a permutation of $1,2,\ldots,n$.
Take the product of the signs of these $2n$ permutations and call it the sign of the latin square. Let $EVEN(n)$ be the number of latin squares with sign $+1$ and let $ODD(n)$ be the number of latin squares with sign $-1$. The conjecture says:
If $n$ is even then $EVEN(n)\neq ODD(n)$.
The original reference is: N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992), 125-134. See also this preprint by Landsberg and Kumar for a recent update.
A: Grundy's game is a two-player mathematical game of strategy.

The starting configuration is a single heap of objects, and the two players take turn splitting a single heap into two heaps of different sizes. The game ends when only heaps of size two and smaller remain, none of which can be split unequally.

Whether the sequence of nim-values of Grundy's game ever becomes periodic is an unsolved problem. Elwyn Berlekamp, John Horton Conway and Richard Guy have conjectured that the sequence does become periodic eventually, but despite the calculation of the first $2^{35}$ values by Achim Flammenkamp, the question has not been resolved.
A: Ore's odd Harmonic number conjecture.
A: Denote by $\sigma(m)$ be the sum of divisors of $m$. When is $ \sigma(n!-1) $ a perfect square? 
