The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
Showing all 993 problems · surprise me
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Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic…
Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
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Square packing in a square
what is the asymptotic growth rate of wasted space?
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Ulam's packing conjecture about the identity of the worst-packing convex solid
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The Tammes problem for numbers of nodes greater than 14 (except 24).
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For which dimensions of sphere differential Poincaré conjecture is true?
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The spherical Bernstein's problem, a generalization of Bernstein's problem
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Carathéodory conjecture
any convex, closed, and twice-differentiable surface in three-dimensional Euclidean space admits at least two umbilical points.
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Cartan–Hadamard conjecture
can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds?
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Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes.
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Chern's conjecture for hypersurfaces in spheres, a number of closely related conjectures.
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Closed curve problem
find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.
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The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a…
The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length
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The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds
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Osserman conjecture
that every Osserman manifold is either flat or locally isometric to a rank-one symmetric space
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Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of…
Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of S^{n+1} is n.
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The big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point…
The big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point sets
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The Dirac–Motzkin conjecture on the minimum number of ordinary lines for n points in the Euclidean plane
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The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies
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Solving the happy ending problem for arbitrary n
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Improving lower and upper bounds for the Heilbronn triangle problem.
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Kalai's 3d conjecture on the least possible number of faces of centrally symmetric polytopes.
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The Kobon triangle problem on triangles in line arrangements
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The Kusner conjecture
at most 2d points can be equidistant in L^1 spaces
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The McMullen problem on projectively transforming sets of points into convex position
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Opaque forest problem on finding opaque sets for various planar shapes
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Orchard-planting problem on the maximum number of 3-point lines attainable by a configuration of n points in the plane
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How many unit distances can be determined by a set of n points in the Euclidean plane?
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Finding matching upper and lower bounds for k-sets and halving lines
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Tripod packing
how many tripods can have their apexes packed into a given cube?
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The Atiyah conjecture on configurations on the invertibility of a certain n-by-n matrix depending on n points in \mathbb{R}^{3}
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Data last refreshed 2026-10-20 03:01 UTC.