Not especially famous, long-open problems which anyone can understand

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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 $0$?

Answer 6

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.

Answer 7

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 $$L(n)=n^{n-2}$$

Answer 8

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).

Answer 9

What is the largest possible volume of the convex hull of a space curve having unit length?

Answer 10

Is there a rectangle that can be cut into $3$ congruent connected non-rectangular parts?

Answer 11

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 $n\geq 5$.

Answer 12

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 $1/2$.)

Answer 13

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.

Answer 14

Is there a positive integer which is both triangular and factorial except these obvious examples: $1, 6, 120$? (Tomaszewski conjecture, http://oeis.org/A000217)

Answer 15

Some pages:
Open Problem Garden
The Open Problems Project edited by Erik D. Demaine, Joseph S. B. Mitchell, Joseph O’Rourke

Answer 16

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).

Answer 17

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 $0$.

Answer 18

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

Answer 19

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.

Answer 20

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.)

Answer 21

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.)

Answer 22

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

$$ g(n) = 2^n + \left\lfloor{\left(\frac{3}{2}\right)^n}\right\rfloor - 2. $$

Answer 23

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.)

Answer 24

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.

Answer 25

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?

Answer 26

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.

Answer 27

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.

Answer 28

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.

Answer 29

Is there such $n\in\mathbb{N}$ that ${^n\pi}\in\mathbb{N}$? (see tetration)

Answer 30

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.