Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
Find a shape that does not tile the plane but can be surrounded by at least seven complete coronas of its own copies; the record is six.
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Find a closed form for the hard-square entropy constant κ ≈ 1.5030480825, the growth rate of independent sets in the square grid.
Prove that any 2^{n−2}+1 points in general position in the plane contain a convex n-gon; open from n = 7 on.
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Find positive integers x, y with 0 < |y² − x³| < √x/100, beating the record ratio of about 46.6 in Hall's conjecture, unbroken since 1998.
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Prove that two competing growth processes on the lattice cannot both grow forever when their speeds differ.
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Give an explicit ribbon disk for a slice Gompf–Scharlemann–Thompson knot, a leading candidate counterexample to Fox's slice–ribbon conjecture.
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Quantify groundwater recharge, flow beneath catchment divides and discharge to the sea, and the processes that control them.
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Show that every tree with m edges can be labelled by distinct numbers 0 to m so that the edge differences are exactly 1 to m.
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Prepare the 23-qubit Golay code's logical zero with fewer than 44 CNOT gates, or prove 44 optimal; at depth 7 the record is 45.
Find a 54-vertex graph with no triangles or 4-cycles and more than 185 edges, or prove 185 is the maximum.
Find a first-order method for geodesically convex optimization with query count logarithmic in 1/ε and cheap iterations.
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Predict crop traits such as yield from genotype across environments well enough to design ideal varieties before they are planted.
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Decide whether every polyhedron can be cut open, with cuts anywhere on its surface, and flattened into one non-overlapping piece.
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Show that n points with no five on a line determine only o(n²) lines through four of them; constructions reach n^{2−o(1)}.
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Find out why floods and droughts cluster into rich decades, whether these periods are changing, and what drives them.
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Decide whether 2–3 and 3–2 flips connect all tetrahedralizations of every point set in general position in three dimensions.
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Construct Nikodym sets over finite fields that leave out as many points as possible, or prove they must fill almost all of the space.
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Find a closed form for Feigenbaum's period-doubling constant δ ≈ 4.6692016091, known to 10,000 digits but not proven irrational.
Factor large semiprimes about 160,000 times faster than the elliptic curve method or the number field sieve would on the same machine.
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Measure how many species are committed to delayed extinction by past habitat loss and fragmentation, and how long until they are lost.
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Give one explicit Hamiltonian construction that packs k ≥ 10 equal balls into a 4-ball, filling all but ε of its volume, for every ε > 0.
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Construct explicit graphs on n vertices with no clique or independent set of size c·log n, matching Erdős's random graphs.
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Find k−1 positive k-th powers summing to a k-th power for some k ≥ 6, extending the counterexamples to Euler's conjecture for k = 4 and 5.
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Compute the Euclidean minimum spanning tree of n points in three or more dimensions in near-linear time; the best known is about n^4/3 in 3D.
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Find a 381-nt RNA sequence that folds into the Eterna100-V2 target Teslagon under ViennaRNA 2.6.4; the players' solution is withheld.
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Find, by algorithm alone, a 282-nt RNA sequence that folds into the Eterna100-V2 target Taraxacum officinale under ViennaRNA 2.6.4.
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Find, by algorithm alone, a 370-nt RNA sequence that folds into the Eterna100-V2 target Shooting Star under ViennaRNA 2.6.4.
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Find a 400-nt RNA sequence that folds into the original Eterna100 Hoglafractal under ViennaRNA 2, or prove that none exists.
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.
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