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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Bellman's lost-in-a-forest problem
find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation
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Borromean rings
are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?
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Connelly’s blooming conjecture
Does every net of a convex polyhedron have a blooming?
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Danzer's problem and Conway's dead fly problem
do Danzer sets of bounded density or bounded separation exist?
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Dissection into orthoschemes
is it possible for simplices of every dimension?
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Ehrhart's volume conjecture
a convex body K in n dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than (n+1)^{n}/n!
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Falconer's conjecture
sets of Hausdorff dimension greater than d/2 in \mathbb{R}^d must have a distance set of nonzero Lebesgue measure
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The values of the Hermite constants for dimensions other than 1–8 and 24
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What is the lowest number of faces possible for a holyhedron?
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Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem
does every Jordan curve have an inscribed square?
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The Kakeya conjecture
do n-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to n?
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The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure…
The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem
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Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one
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Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.
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In Meissner body)
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Moser's worm problem
what is the smallest area of a shape that can cover every unit-length curve in the plane?
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The moving sofa problem
what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
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In parallelohedron
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Ropelength problems
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Does every convex polyhedron have Rupert's property?
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Shephard's problem (a.k.a. Dürer's conjecture)
does every convex polyhedron have a net, or simple edge-unfolding?
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Is there a non-convex polyhedron without self-intersections with more than seven faces, each of which shares exactly one edge with each…
Is there a non-convex polyhedron without self-intersections with more than seven faces, each of which shares exactly one edge with each other?
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The Thomson problem
what is the minimum energy configuration of n mutually-repelling particles on a unit sphere?
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Convex uniform 5-polytopes
find and classify the complete set of these shapes
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Hilbert's third problem for non-Euclidean geometries
in spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?
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Babai's problem
which groups are Babai invariant groups?
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Brouwer's conjecture on upper bounds for sums of eigenvalues of Laplacians of graphs in terms of their number of edges
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γ–θ conjecture
Does there exist a graph G such that the dominating number, eternal domination number, and clique covering number satisfy \gamma(G)=\gamma_\infty(G)
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Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs
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Meyniel's conjecture that cop number is O(\sqrt n)
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Data last refreshed 2026-10-20 03:01 UTC.