Hadwiger’s illumination conjecture
Can every convex body in n dimensions have its entire boundary illuminated from at most 2ⁿ directions?
Mathematics
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Symmetric Mahler conjecture
Do cubes and their dual shapes minimize a symmetric body’s volume multiplied by its polar’s volume?
Kissing number in five dimensions
How many equal five-dimensional balls can touch one central ball without overlapping each other?
Zariski cancellation for affine three-space
If adding one affine coordinate makes a complex variety look like four-dimensional affine space, was it already affine three-space?
Borel conjecture
For certain closed manifolds, does matching their homotopy structure force them to be the same topological shape?
Novikov conjecture
Do a manifold’s higher signatures stay unchanged when its homotopy structure is preserved?
Brocard–Ramanujan problem
Can a factorial plus one be a perfect square beyond the three known examples?
Singmaster’s conjecture
Is there a fixed limit on how often any number greater than one can appear in Pascal’s triangle?
Erdős–Ulam problem
Can points be dense throughout the plane while every distance between two of them is rational?
Moser’s convex worm problem
What is the smallest-area convex region that can contain every planar curve of length one after repositioning it?
Minimum opaque barrier for a square
How little total barrier length can intercept every straight line that passes through a unit square?
Seymour’s second-neighborhood conjecture
Must a directed graph without two-way edges have a vertex reaching at least as many new vertices in two steps as in one?
Borsuk’s conjecture in dimension four
Can every bounded four-dimensional set be split into five pieces of strictly smaller diameter?
Kaplansky’s zero-divisor conjecture
Can two nonzero finite algebraic combinations of elements of a torsion-free group multiply to zero?
Aaronson–Ambainis influence conjecture
Must a bounded low-degree polynomial that varies appreciably have an input bit with noticeable influence?
Deterministic prime construction
Can an algorithm reliably construct a prime of any requested bit length in polynomial time without randomness?
NP versus coNP
Whenever yes answers have short efficiently checkable certificates, must no answers have them too?
Small-Set Expansion hypothesis
Is it computationally hard to distinguish a graph with a small nearly isolated set from one where every equally small set strongly expands?
Optimal propositional proof system
Can one efficiently checkable proof system match every other system’s proof lengths within a polynomial factor?
Densest-k-subgraph approximation
Can a fast algorithm always choose k vertices spanning within a fixed factor of the largest possible number of edges?
Polynomial-time parity-game solving
Can the winner of every finite parity game be determined in polynomial time?
Conway’s thrackle conjecture
Can a planar thrackle have more edges than vertices?
Gyárfás–Sumner conjecture
Does excluding any fixed induced tree control a graph’s coloring needs once its largest clique is bounded?
Barnette’s Hamiltonian conjecture
Does every cubic, bipartite, planar, 3-connected graph contain a cycle visiting every vertex exactly once?