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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Explain why the test loss of neural networks follows power laws in model size, data and compute, and predict the exponents from properties of the data and architecture. Reproducible small-scale experiments serve as evidence.
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.
Prove halting or non-halting of specific small Turing machines that current deciders cannot resolve.
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.
Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
Close `sorry`s in Lean formalisations of Erdős problems, prove special cases, or formalise known partial results.
Assign biological function to the genes of the minimal synthetic cell JCVI-syn3A that are still uncharacterised, several of which are essential for growth.
Construct Hadamard matrices for orders 4k where none is known, starting with the smallest open orders.
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
Find large subsets of F_3^n with no three points on a line (no x, y, z distinct with x + y + z = 0).
Explain and overcome the degradation of metal-halide perovskite solar cells so that their high efficiencies persist for decades under real operating conditions.
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.
Improve upper bounds C(v,k,t) for covering designs listed in the La Jolla Covering Repository.
Find sorting networks with fewer comparators than the best known for n ≥ 13 inputs, or prove optimality.
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 a bilinear algorithm multiplying 3×3 matrices with fewer than 23 multiplications, or raise the lower bound.
Find bilinear algorithms that multiply two 4×4 matrices with fewer multiplications. The records are 48 over Q and C (2025) and 47 over GF(2) (2022).
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.
Determine which source populations and emission mechanisms produce fast radio bursts, and whether repeating and apparently non-repeating bursts share a common origin, using public CHIME/FRB and other catalogues.
Narrow the gap 36 ≤ R(4,6) ≤ 40. A 2-colouring of K_36 with no red K_4 and no blue K_6 would raise the lower bound; lowering the upper bound needs reproducible exhaustive computation.
Narrow the gap between the known lower and upper bounds for R(5,5), currently 43 ≤ R(5,5) ≤ 46.
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.
Determine whether marine ice-sheet and ice-cliff instabilities can drive rapid Antarctic retreat this century, and how much they widen sea-level projections.
Find better polynomial-time approximation algorithms for the metric Traveling Salesman Problem and prove the conjectured 4/3 integrality gap of the subtour LP. The best known ratio is 3/2 − ε with ε > 10^−36 (Karlin–Klein–Oveis Gharan).
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.
