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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.

http://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.

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    $\begingroup$ You might get more success if you sampled certain open problem lists and indicated which ones fit your list and which ones did not. I could mention various combinatorial problems such as integer complexity, determinant spectrum, covering design optimization, but I can't tell from your description if they would be suitable for you. Gerhard "They Are Suitable For Me" Paseman, 2012.06.21 $\endgroup$ Jun 21 '12 at 19:11
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    $\begingroup$ Here is some collection of some other "collect open problems" quests. on MO: mathoverflow.net/questions/96202/… PS Nice question ! PSPS may be add tag "open-problems" $\endgroup$ Jun 21 '12 at 20:53
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    $\begingroup$ To save the search for explanation of cryptic acronyms for those of us outside US, K-12 means high school. @Mahmud: You are using a wrong meaning of the word “problem”. The TSP is not an unproved mathematical statement, it is a computational task. $\endgroup$ Jun 22 '12 at 12:05
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    $\begingroup$ More precisely, K-12 means anything up to high school (K = Kindergarten, 12 = 12th grade, and K-12 covers this range). $\endgroup$
    – Henry Cohn
    Jun 22 '12 at 13:05
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    $\begingroup$ There seems to be a claimed proof of the union-closed sets conjecture by Blinovsky arxiv.org/abs/1507.01270 $\endgroup$
    – Marco
    Oct 22 '15 at 14:08

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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).

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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.

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    $\begingroup$ Surely you don’t meen Eulerian? For example, $G_4$ isn’t Eulerian as it has many odd degree vertices. $\endgroup$
    – OHO
    Mar 3 '20 at 7:36
  • $\begingroup$ I meant "Hamiltonian". Thanks for pointing that out. $\endgroup$ Mar 23 '20 at 21:53
  • $\begingroup$ I can't find it in Guy's book. A recent paper is Chen, Fu, and Guo, From a consequence of Bertrand's postulate to Hamiltonian cycles, arxiv.org/abs/1804.07104 The authors prove the conjecture holds for infinitely many $n$. They seem to be unaware of previous discussions of the conjecture. $\endgroup$ Mar 23 '20 at 23:47
  • $\begingroup$ Also, the question has been discussed here on MO: mathoverflow.net/questions/241569/… where Zhi-Wei Sun gives the reference Antonio Filz [J. Recreational Math. 14(1982), p.64] and Max Alekseyev gives the reference oeis.org/A051252 – oh, and it is in Guy's book, in the section (C1) on Goldbach's conjecture, with the attribution to Filz. $\endgroup$ Mar 23 '20 at 23:59
  • $\begingroup$ Another reference is Stan Wagon's Problem of the week 1218, mathforum.org/wagon/current_solutions/s1218.html $\endgroup$ Mar 24 '20 at 0:11
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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.

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Does every nonseparating planar continuum have the fixed point property?

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    $\begingroup$ For those of us whose general topology is rusty: what is the K-12 level formulation of this question? $\endgroup$
    – Yemon Choi
    Jun 21 '12 at 22:50
  • $\begingroup$ I'm happy to learn that this unknown, even if doesn't meet the stipulations. So it got my up vote. $\endgroup$ Jun 22 '12 at 3:02
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    $\begingroup$ Uniform continuity is a K-12 concept now? $\endgroup$ Jun 22 '12 at 14:25
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    $\begingroup$ Thanks Yemon. I agree with Doug that uniform continuity is not a stand alone K-12 concept. I mentioned it in order to avoid trying to decode its meaning in K-12 terms in this context, as follows: $\endgroup$
    – Paul Fabel
    Jun 22 '12 at 17:56
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    $\begingroup$ f is a set of 4-tuples (x1,y1,x2,y2) so that each point (x1,y,1) in K appears precisely once as the 1st two coordinates of some point in f, and we also require that the last two coordinates of each point of f is a point of K. For each positive radius R we can find a smaller positive radius r(R) so that if (x1,y1,x2,y2) is in f, then if (w1,z1) is both in K and also in the disk of radius r(R) centered at (x1,y1), and if (w1,z1,w2,z2) is in f, then (w2,z2) is in the disk of radius R centered at (x2,y2). $\endgroup$
    – Paul Fabel
    Jun 22 '12 at 17:56
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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 $V = 9/2$.

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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?

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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?

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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).

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Are there infinitely many partition numbers divisible by $3$? See A000041.

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  • $\begingroup$ Could you perhaps expand a bit, i.e. for example what makes this a difficult problem, and what is known about the same question for divisors other than $3$? $\endgroup$
    – Stefan Kohl
    Nov 9 '15 at 10:12
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    $\begingroup$ For example, it is known that there are infinitely many divisible by 2. It is hard for me to estimate how difficult the problem is, but it was publicly raised at OEIS more than 10 years ago (and I'm sure it had been discussed before), it is still open, and it is about a famous sequence that received much attention (and new interesting results) during last decade. The sequence graph suggests that the distribution of such numbers is quite irregular, they do not seem to to lie nicely on a smooth curve. $\endgroup$ Nov 9 '15 at 17:23
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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.

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(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.

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    $\begingroup$ It was proved to be true if q is not a prime or if q is prime and n is bounded by a function of q. $\endgroup$ Dec 8 '17 at 12:04
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Can a square with integer sides contain a point whose distance from each corner is an integer?

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    $\begingroup$ D19 in Guy, Unsolved Problems in Number Theory, with many references. $\endgroup$ Mar 24 '20 at 0:14
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    $\begingroup$ Is this Belk's problem above, June 2015? $\endgroup$
    – Ben McKay
    Jun 24 '20 at 10:20
  • $\begingroup$ Yes, it is. (I just noticed Belk's contribution.) $\endgroup$ Jun 24 '20 at 17:54
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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.

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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? ;)

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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 ) describes some current approaches.

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    $\begingroup$ You seem to allude that it has not been resolved yet. But in that case you should explain what you mean by that. On the other hand, the continuum hypothesis is so well-known that it does not fit to the question. $\endgroup$ Jul 19 '12 at 12:17
  • $\begingroup$ +1 because the note you linked is really interesting! Although the "problem" is rather open-ended, I think it could be stated in a more well-defined way. Here's a stab at it: "If we want to settle the continuum hypothesis, which axioms should we add to ZFC in order to do it?" $\endgroup$
    – Vectornaut
    Jul 22 '12 at 18:24
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    $\begingroup$ By the way, I think your statement that "everyone thinks it's resolved" is a little misleading. Maybe it would be better to say it this way: "most people think there's nothing left to say about it, because it's been proven independent of ZFC." $\endgroup$
    – Vectornaut
    Jul 22 '12 at 18:26
  • $\begingroup$ this is not an open problem. It has been settled (like many impossibility results) a long time ago. $\endgroup$ Sep 10 '19 at 4:08
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N. M. Katz: "Simple Things we don't know": https://web.math.princeton.edu/~nmk/pisa16.pdf

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  • $\begingroup$ These will be a better fit here: mathoverflow.net/questions/101169/… when the collective net gods deem it time to reopen the question. $\endgroup$ Jul 14 '12 at 19:07
  • $\begingroup$ These will be a better fit here: mathoverflow.net/questions/101169/ when the collective net gods deem it time to reopen the question. Every vote helps, thanks. $\endgroup$ Jul 14 '12 at 19:08
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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?

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  • $\begingroup$ Is it even clear that there's a lower bound in the non-convex case? I feel like if I had to bet on a number I'd probably pick 0 by analogy with Kakeya... $\endgroup$ Aug 18 at 19:32
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Ore's odd Harmonic number conjecture.

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    $\begingroup$ I suspect this fails on the fame criterion. Gerhard "Ask Me About System Design" Paseman, 2012.06.27 $\endgroup$ Jun 27 '12 at 14:22
  • $\begingroup$ Yes, but Ore's odd harmonic number conjecture, which if true, implies no odd perfect numbers, seems just right here. $\endgroup$ Jun 28 '12 at 4:35
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    $\begingroup$ Well, now it's so obscure that it requires a context/explanation. $\endgroup$ Jan 6 '14 at 20:45
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Denote by $\sigma(m)$ be the sum of divisors of $m$. When is $ \sigma(n!-1) $ a perfect square?

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    $\begingroup$ Is this a long-open problem? Does it have a history? $\endgroup$ Sep 15 '17 at 23:02
  • $\begingroup$ Numbers k such that σ(k) is a square are tabulated at oeis.org/A006532 – it is not even known that there are infinitely many of them (although this would follow from standard conjectures) $\endgroup$ Oct 17 '17 at 19:17
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    $\begingroup$ Isn't jstor.org/stable/10.4169/… proving that there are infinitely many? $\endgroup$ Jan 12 '19 at 14:23
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Does the decimal expansion of $2^n$ contain the digit $7$ for all sufficiently large $n$?

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  • $\begingroup$ Is there anything specific about base 10 or digit 7? $\endgroup$ Aug 19 at 16:14
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