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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 (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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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 –  Gerhard Paseman Jun 21 '12 at 19:11
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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" –  Alexander Chervov Jun 21 '12 at 20:53
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Nice question!! –  Suvrit Jun 22 '12 at 3:25
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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. –  Emil Jeřábek Jun 22 '12 at 12:05
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More precisely, K-12 means anything up to high school (K = Kindergarten, 12 = 12th grade, and K-12 covers this range). –  Henry Cohn Jun 22 '12 at 13:05

96 Answers 96

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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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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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 $$ \nabla \cdot (|\nabla u|^{p-2}\nabla u) = 0 \ \ \text{in}\ \Omega, $$ 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.

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Ore's odd Harmonic number conjecture

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Well, now it's so obscure that it requires a context/explanation. –  Victor Protsak Jan 6 '14 at 20:45

Is there a triangle that can be cut into $7$ congruent triangles? (no)

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Nice problems but what are the sources? –  Alexander Chervov Jul 24 '12 at 20:23
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I heard this in a personal communication. But it turns out this is already settled negatively in 2008: michaelbeeson.com/research/papers/SevenTriangles.pdf –  Vladimir Reshetnikov Jul 28 '12 at 20:34
  • Is the Ring of Periods actually a field? (most likely, no)
  • Is the equality of periods decidable? (hopefully, yes)
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problems which anyone can understand ? Uhhh –  Denis Serre Sep 25 '12 at 7:50

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