Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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Determine how much of the atmospheric methane increase since 2007 comes from wetlands, fossil sources and agriculture versus a weakening sink, using public observations and reproducible inversions.
Map the boundary of classical simulability for quantum-advantage random circuit sampling experiments by improving tensor-network and other classical algorithms, with reproducible cost estimates and fidelity benchmarks.
Design enzymes from scratch that catalyse a chosen benign chemical transformation with natural-enzyme efficiency, and predict activity well enough that most designs work.
Predict thermoelectric figure of merit zT from first principles or data well enough to find new high-performance, earth-abundant thermoelectric materials.
Find a Costas array of order 32 or 33, the smallest orders for which none is known, or extend the complete enumeration of Costas arrays beyond order 29.
Predict, from a molecular diagram alone, which crystal forms a compound adopts and their relative stability — the task probed by the CCDC blind tests.
Establish whether El Nino-Southern Oscillation events can be predicted skilfully at lead times beyond about a year, with skill demonstrated in fair, reproducible hindcasts.
Explain why some networks generalise long after fitting their training data, and predict when this happens. Reproducible small-model experiments serve as evidence, e.g. modular arithmetic transformers whose circuits can be reverse-engineered.
Find the shortest Golomb ruler (all pairwise mark differences distinct) with n marks. Optimality is proven up to 28 marks (length 585, distributed.net, 2022); 29 marks is the first open case, and shorter rulers for larger n would beat long-standing constructions.
For each n, find the largest radius r such that n non-overlapping circles of radius r fit in a unit square. Optimality is proven only for small n. For larger n, improve the best known packings or prove new cases optimal.
Discover and certify new periodic orbits of the Newtonian three-body problem (planar and three-dimensional, equal and unequal masses), with reproducible initial conditions and error control.
Sharpen the rigorous numerical bounds on the scaling dimensions Δσ and Δε of the 3D Ising conformal field theory, and reconcile them with Monte Carlo estimates.
Predict room-temperature ionic conductivity of solid lithium-ion conductors from structure and composition, and propose new candidates that beat known sulfides on conductivity plus stability.
Predict from composition alone whether an alloy forms a bulk metallic glass and how thick it can be cast, and use this to find new glass formers.
Go beyond single-structure prediction — predict the alternative conformations and equilibrium populations that proteins actually adopt, and validate against public simulation and experimental data.
Predict binding affinities for protein-ligand complexes accurately enough to be useful prospectively, and show it on benchmarks that are free of train-test leakage.
Settle open finite combinatorial questions with SAT solvers that emit checkable unsatisfiability proofs (DRAT/LRAT). Examples of solved cases are Boolean Pythagorean triples, Schur number five, Keller's conjecture in dimension 7, and the empty hexagon number.
Identify rare-earth-free compounds with enough magnetization, magnetocrystalline anisotropy and Curie temperature to fill the performance gap between ferrites and Nd-Fe-B magnets.
Predict and control which product a CO2-reduction catalyst makes — especially C2+ products such as ethylene and ethanol — from computable descriptors rather than trial and error.
Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.
Find the longest induced path (snake) in the n-dimensional hypercube Q_n. Optimal lengths are known only up to n = 8 (98); for n = 9–13 new records were set in 2026 and further improvements are open.
Find the largest graphs with maximum degree d and diameter k. Records for 3 ≤ d ≤ 20 and 2 ≤ k ≤ 10 are tabulated and mostly far below the Moore bound; whether a Moore graph of degree 57 (3250 vertices) exists is a famous open case.
Combine public NICER pulse-profile posteriors, gravitational-wave tidal-deformability constraints and pulsar masses into reproducible, well-calibrated constraints on the neutron-star equation of state.
Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.
Find configurations of n unit point charges on the sphere that minimise Coulomb energy. Global optimality is proven only for a few n, so the tasks are to lower best known energies and to prove new cases optimal.
Improve reproducible, circuit-level-noise thresholds and logical error rates for surface codes and quantum LDPC codes through better codes, syndrome circuits and decoders, benchmarked with open simulators such as Stim.
Find a particulate photocatalyst that splits water with high quantum efficiency under visible light, closing the gap between near-perfect UV performance and the low solar-to-hydrogen efficiency of real panels.
Build a connectome-constrained model of the C. elegans nervous system that reproduces measured neural activity and behaviour, and validate it against public imaging and connectome data.
Establish, check and extend rigorous computer-assisted proofs that smooth solutions of the 3D incompressible Euler equations (and related models) develop singularities in finite time.
Prove Hill's conjecture cr(K_n) = ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋ and Zarankiewicz's conjecture for K_{m,n}. Exact values are known only for small cases (K_n up to n = 14, K_{m,n} for m ≤ 6 and a few m = 7 cases).
Construct an exchange-correlation approximation that reaches chemical accuracy (~1 kcal/mol) across broad main-group chemistry at semi-local cost, and show it on public benchmarks.
Decide whether an odd perfect number exists. Any such number exceeds 10^1500 and has at least 10 distinct prime factors; progress tightens these constraints.
Prove that a symmetric informationally complete POVM (d^2 equiangular lines in C^d) exists in every dimension d, and extend the list of dimensions with exact or numerical solutions.
Determine reliably, with controlled numerics, where the doped two-dimensional Hubbard model (with and without next-nearest-neighbour hopping t′) is superconducting, striped or otherwise ordered.
Resolve the disagreement between lattice-QCD and data-driven (e+e− → hadrons) evaluations of the leading hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.
Establish which high-pressure hydride superconductivity claims are robust and how accurately ab initio electron-phonon theory predicts their critical temperatures.
Decide whether four (or seven) mutually unbiased bases exist in C^6; only three are known, and a complete set of seven is widely believed not to exist.
Narrow the uncertainty in equilibrium climate sensitivity — the long-term warming for a doubling of CO2 — using reproducible analyses of public model output and observational records.
Predict three-dimensional RNA structures from sequence with accuracy comparable to protein structure prediction, including targets for which no structural template exists.
Build earthquake forecast models whose skill is demonstrated in prospective, pre-registered tests such as those run by CSEP, and quantify how much predictability exists at all.
Prove that every even integer greater than 2 is the sum of two primes. It has been verified up to 4·10^18, and the ternary (odd) version was proved by Helfgott.
Prove that every synchronizing complete DFA with n states has a reset word of length at most (n−1)². The best general upper bound is about 0.1654·n³ (Shitov 2019). The conjecture has been verified by computer for small automata.
Determine how many colours are needed so that no two points of the plane at distance exactly 1 share a colour. The answer is known to be 5, 6 or 7; a concrete sub-goal is a smaller 5-chromatic unit distance graph than the 509-vertex record.
Prove that iterating n ↦ n/2 (n even), 3n + 1 (n odd) reaches 1 from every positive integer. It has been verified up to 2^71, and Tao showed that almost all orbits attain almost bounded values.
Prove that 4/n = 1/x + 1/y + 1/z has a solution in positive integers for every n ≥ 2. It has been verified to at least 10^17, and all n outside a few residue classes are covered by explicit identities.
Is every set of 2^{n−2}+1 points in general position in the plane guaranteed to contain n points in convex position? Known exactly up to n = 6 (17 points); the first open case is whether 33 points force a convex 7-gon.
Every tree with n vertices has a graceful labelling, i.e. vertex labels 0..n−1 whose edge differences are exactly 1..n−1. It has been verified for all trees with at most 35 vertices; extending this range and proving new classes graceful are open.
Explain why local distance-ladder measurements of the Hubble constant H0 disagree with the value inferred from the cosmic microwave background under ΛCDM, or show that the disagreement is systematic.
Decide whether every finite group occurs as the Galois group of a Galois extension of Q. All sporadic groups are now realised (M23 in 2026); most transitive groups of degree 24 are not yet.
For k+1 runners with distinct constant speeds on a unit circular track, each runner is at some time at distance at least 1/(k+1) from all others. Computer-assisted proofs now cover up to 13 runners; the general case is open.
