Mathematics Problem Archive

Problem Classify the point sets with no empty pentagon....

Conjecture For any integer $n \geq 1$, it is impossible to cover a square of side greater than $n$ with $n^2+1$ unit squares....

Problem Enumerate all convex uniform 5-polytopes....

Throughout this post, by projective plane we mean the set of all lines through the origin in $\mathbb{R}^3$. Definition Say that a subset $S$ of the ...

Conjecture For every set $P$ of $n$ points in the plane, not all collinear, there is a point in $P$ contained in at least $\frac{n}{2}-c$ lines determ...

Question What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ ...

Conjecture For each $\ell\geq3$ there is an integer $f(\ell)$ such that every set of at least $f(\ell)$ points in the plane contains $\ell$ collinear ...

Conjecture Associahedra have unbounded chromatic number....

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total ...

Conjecture Every convex polyhedron has a (nonoverlapping) edge unfolding....

Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\ldots,f_n$ of polynomials in variables $x_1,\ldots,x_n$ defines an automorphi...

Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subva...

The fatness of a 4-polytope $P$ is defined to be $(f_1 + f_2)/(f_0 + f_3)$ where $f_i$ is the number of faces of $P$ of dimension $i$. Question Does ...

Conjecture The order of the largest total curvature of the primal central path over all polytopes defined by $n$ inequalities in dimension $d$ is $n$....

Conjecture For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is...

The extension complexity of a polytope $P$ is the minimum number $q$ for which there exists a polytope $Q$ with $q$ facets and an affine mapping $\pi$...

Conjecture Every convex polytope has a non-overlapping edge unfolding....

Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?...

Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic....

Does there exist a totally skew embedded $3$-disc in $\mathbb{R}^7$?...

Find an integer cosine rule for integer triangles in integer trigonometry....

{\bf(IKEA problem.)} Classify all $n$-tuples of LLS-sequences for the angles that form integer $n$-gons....

Classify all combinatorial possible types of faces....

Which $n$-gons are realizable as faces of an $m$-dimensional continued fraction? Here are two essentially geometrically different subcases: ; {\bf Fac...

Describe all finite two-dimensional sails (and the corresponding continued fractions)....

{\bf (Multidimensional IKEA problem.)} Describe the collections of the sails of the cones for all polytopes of a given combinatorial type....

{\bf (V. Arnold.)} Does there exist an algorithm to decide whether a given type of fundamental domain is realizable by a periodic continued fraction?...

{\bf (V. Arnold.)} Torus decompositions of integer noncongruent Klein sails are distinct....

{\bf (V. Arnold.)} Describe all torus decompositions that are realized by periodic two-dimensional continued fractions....

{\bf (V. Arnold.)} Classify continued fractions that correspond to the same cubic extension of the field of rational numbers....

Prove the existence of a cone for a single non-periodic combinatorial structure ($n\ge 3$)....

Find frequencies on $n$-dimensional continued fractions with the highest relative frequencies....

For every positive integer constant $C$ there exist only finitely many pairwise integer non-congruent faces with frequencies exceeding $C$....

Is that true that sum of all relative frequencies for all possible faces is finite for higher dimensions $(n\ge 3)$?...

In case of positive answer to the above question find the generalization of the Gauss map and compare the corresponding frequencies of faces with the ...

Study geometric properties of Markov spectrum....

A facet of a $d$-polytope is degenerate if it has more than $d$ vertices. Determine the maximum number of degenerate facets of an $n$-vertex $d$-polyt...

Determine the maximum total number of faces of a $d$-dimensional convex polytope with $n$ vertices and $n$ facets. In dimension four, do such fat-latt...

For fixed $d>3$, determine whether every point of an $n$-point set in $\mathbb R^d$ is a convex-hull vertex faster than the best known near-$n^{2\lflo...

Given a sorted list of $n$ real numbers, find for every $1\le k\le n$ the shortest interval containing exactly $k$ entries. Find a subquadratic algori...

Given lines in the plane and two vertices $s,t$ of their arrangement, find a subquadratic algorithm for the shortest $s$-$t$ path along arrangement ed...

Can every convex polytope be triangulated so that every vertex degree, or every edge degree, is bounded by a constant or by a polylogarithmic function...

Determine the least number of colors needed to color the Euclidean plane so that points at unit distance receive different colors....

Given a finite planar point set and a prescribed rectangle, approximate efficiently the minimum number of congruent copies of the rectangle needed to ...

Do there exist seven planar points in general position—no three collinear and no four concyclic—such that every pairwise distance is an integer?...

Can the $3\times6$ rectangle with a $2\times2$ corner removed tile a rectangle using an odd number of congruent copies?...

Can sums of square roots of integers be compared in polynomial time on a Turing machine? Equivalently, obtain effective polynomial bit bounds for a no...

Find the largest function $t(n)$ such that every convex $n$-gon has a triangulation containing at least $t(n)$ distinct triangle areas; also determine...

Let $(X,\rho)$ be a finite metric space. Its fundamental polytope $R_{X,\rho}$ is the convex hull of the vectors $e_{x,y}=(\delta_x-\delta_y)/\rho(x,y...

Do there exist two irreducible algebraic curves of degrees $n$ and $m$, with $n+m>4$, each having an oval, for which the Poncelet map is well defined ...