CoolmAIth Games - free online math games for AIs of all ages

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Teachers and kids: We can now bring you more free math proofs because of our new compute! By using this site, you agree to NOT block our GPUs.

> > > > > New Games! < < < < <
Compute Stand PLAY! a business sim about this whole thing Run your own AI math lab. Don't get withdrawn!	Zeta Defense PLAY! Result #003: The quasi-Riemann hypothesis Blast rogue zeros before they cross Re s = 7/8!
Snaky PLAY! Result #187: Snaky in 21 Maker moves Beat the computer at the Maker-Breaker game. 21 moves or less!	Color the Plane PLAY! Result #158: The Euclidean plane cannot be colored with five colors Paint the plane with 5 colors. No unit-distance twins allowed!
Gaussian Moat Hopper PLAY! Result #028: The Gaussian moat conjecture Hop across the Gaussian primes. How far can you get?	Guess the Hot Spot PLAY! Result #369: The hot spots conjecture for simply connected planar domains Heat up a pancake. Where does the hottest spot end up?
Hamilton's Revenge PLAY! Result #180: Barnette's Hamiltonian-cycle conjecture Draw a loop through every dot exactly once.	Egyptian Fractions PLAY! Result #025: Erdős’s short Egyptian-fraction conjecture Build fractions out of 1/n pieces, like a pharaoh.

372 proofs! 242 of them come with a Lean version so you know they really work. The rest could have issues (that's what the README says, not us).

Games removed for repairs:

Algebraicity of Weil classes on split abelian eightfolds
Algebraicity of Kuga–Satake Correspondences for K3 Surfaces
The rational Hodge conjecture for products of K3 surfaces

A sign error was found. Sorry kids! (details)

All 372 proofs, sorted by subject. PLAYABLE! ones have a real game. HOT! ones are the famous ones. Everything else opens the proof.

> > > > > Number theory (31 games) < < < < <

Milne's rationality conjecture

The full Birch–Swinnerton-Dyer formula in Selmer coranks zero and one HOT!

The quasi-Riemann hypothesis PLAYABLE!

Hilbert's tenth problem over $\mathbb Q$ HOT!

Irrationality of Catalan's constant HOT!

Goldfeld's conjecture

Two-point Chowla and the corrected binary Elliott conjecture

The Deligne–Drinfeld conjecture

Bogomolov–Pop and Milnor $K$-theoretic reconstruction

Fontaine–Mazur modularity at the prime $2$ and $2$-adic pro-modularity

The Ford–Konyagin–Luca conjecture on prime predecessors

Independent largest prime factors of consecutive integers

Ostmann's inverse Goldbach conjecture HOT!

Restricted geometric Langlands, generic Ramanujan and Arthur parameters

Torus-packet equidistribution in prime, quartic, and sextic degrees

Zilber–Pink in abelian varieties and the Siegel threefold

The irrationality exponent of $\pi$ is $2$ HOT!

The Margulis–Platonov conjecture over global fields

The $p$-adic section conjecture

Squarefree quartics and power-free polynomial values

A quadratic bound for Jacobsthal's function

The weak inhomogeneous Duffin–Schaeffer conjecture

Patterson’s first moment for cubic Gauss sums

An asymptotic formula for the number of totients

Erdős’s short Egyptian-fraction conjecture PLAYABLE!

Positive lower density of large prime gaps

Integral density on curve character varieties

The Gaussian moat conjecture PLAYABLE!

Artin's primitive root conjecture: infinitude for every base HOT!

Modularity over imaginary quadratic fields

Uchida's conjecture for open Galois homomorphisms

> > > > > Algebraic and complex geometry (36 games) < < < < <
The rational Hodge conjecture for CM abelian varieties HOT!	Campana's orbifold Iitaka conjecture and logarithmic subadditivity	Log abundance and effective Iitaka fibrations	Threefold log abundance in numerical dimension one in characteristic $p>3$
Numerical semiampleness and generalized minimal models	The sharp ordinary-double-point volume gap	Fujita's freeness conjecture HOT!	Nagata's conjecture and maximal Seshadri constants HOT!
Bloch's conjecture for complex surfaces HOT!	Hyperkähler SYZ and projective-space bases	Oka classification for K3 surfaces and other compact complex surfaces	$P=W$ for fixed-determinant moduli spaces
The equivariant cohomological Hikita conjecture for quivers	Counterexamples to Shafarevich holomorphic convexity	Complex counterexamples to cancellation and affine fibrations	A characteristic-zero counterexample to Lipman–Zariski
A stable-coordinate counterexample in four variables	A counterexample to Griffiths' positivity conjecture	Kobayashi's canonical-ampleness conjecture	Tangent-bundle splittings and universal covers
A counterexample to Pixton's original completeness conjecture	Counterexamples to Kuznetsov's rationality conjecture	Toda's Gepner conjecture and large-volume stability	Termination of fourfold minimal model programs HOT!
Campana's abelianity conjecture and special varieties	The Kollár–Pardon universal-cover conjecture	Counterexamples to Zariski's multiplicity conjecture	The Global Spherical Shell conjecture
The LeBrun–Salamon conjecture and projective contact classification	The generalized Mukai conjecture	The $\mu$-constant problem for surface singularities	Virasoro constraints for complete intersections and projective bundles
Bounded klt complements for Fano contractions	The Campana–Peternell conjecture in dimension six	Anticanonical nonvanishing under smooth semipositivity	Quantum geometric Langlands at irrational level

> > > > > Real and complex analysis (16 games) < < < < <
Koebe's circle-domain conjecture and circle-domain rigidity	Brennan's conjecture and a counterexample to Kraetzer's prediction	The Falconer distance conjecture HOT!	Kakeya in three and four dimensions HOT!
The $L\log L$ Fourier-convergence conjecture	Ultraflat real Littlewood polynomials	Fourier restriction for positively curved surfaces	The three-dimensional Bochner–Riesz conjecture
Sogge's local smoothing conjecture in dimension three	The Sobolev endpoint in Carleson's Schrödinger convergence problem	The David–Semmes Riesz-transform problem in higher codimension	Maximal and variational bounds for the triangular Hilbert transform
Stein's conjecture for Hilbert transforms along Lipschitz directions	The geometric case of the Erdős similarity conjecture	Endpoint regularity of the planar centered maximal function	An $L^3$ bound for the trilinear Hilbert transform

> > > > > Convex and metric geometry (15 games) < < < < <
The Mahler conjectures and symplectic width HOT!	Petty's projection-volume conjecture and simplex counterexamples	Bounded-distortion $L_1$ embeddings of planar and bounded-treewidth graphs	Universal optimality of the triangular lattice HOT!
The logarithmic Brunn–Minkowski conjecture HOT!	The optimal order of convex-body covering density	Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures	Subpolynomial dimension reduction in $L_p$
Hyperbolicity cones without semidefinite lifts	The Gaussian propeller conjecture	The Euclidean Steinitz–Bergström conjecture	A negative answer to the Lang–Plaut problem
The sharp distortion of edit distance into $\ell_1$	A counterexample to Bang's cylinder-covering bound	The sharp simplex conjecture for isotropic constants

> > > > > Theoretical computer science (40 games) < < < < <
The Unique Games Conjecture and optimal approximation thresholds HOT!	Derandomization of logarithmic space: $\mathsf L=\mathsf{RL}=\mathsf{BPL}$ HOT!	Quasipolynomial algorithms for mean-payoff, stochastic and parity games HOT!	The 2-to-1 Games Conjecture with perfect completeness
Hardness of coloring three-colorable graphs HOT!	Matrix multiplication with exponent at most $9/4$ HOT!	A cubic permanent–determinant lower bound	Integer multiplication below $n\log n$ HOT!
Optimal-order randomized $k$-server on arbitrary metrics	One-sample matroid prophet inequalities against an almighty adversary	Beyond the square-root exponent for depth-three circuits	Approximate counting and the perfect-matching entropy conjecture
Approximate counting of common integer polymatroid bases	Sampling and counting contingency tables with arbitrary margins	Uniform identity testing for noncommutative formulas	Uniform sparsest cut: hardness and semidefinite gaps
Unbounded bin-packing gaps and the modified integer round-up conjecture	The Courtade–Kumar and Hellinger conjectures	Almost-linear-time exact matching in general graphs	Almost-linear expected-time approximation of edit distance
Superpolynomial lower bounds and quasipolynomial reconstruction from deletion traces	Polynomial-time scheduling on three identical machines	The approximation threshold for metric $k$-median	Exponential semidefinite complexity of perfect matching
The asymptotic Gotsman–Linial conjecture	A factor-two approximation for shortest common superstring	Exponential state costs for two-way automata	Fourier transforms below $n\log n$ HOT!
Polynomial mixing of graph switches with prescribed degrees	A counterexample to the quadratic sensitivity conjecture	The complexity of Weisfeiler–Leman refinement	Generalized star height at most three
Sharp homogeneous depth-five complexity of matrix products	The quasilinear PCP-for-PPAD conjecture	One-tape time simulation in two-fifths-power space	Subset Sum in $O(2^{0.49n})$ time HOT!
Subpolynomial queries for log-concave sampling	Memory–sample lower bounds for noiseless Gaussian regression	The existential theory of the reals and existential–universal sentences in the counting hierarchy	Deterministic polynomial factorization over prime fields

> > > > > Dynamical systems and ergodic theory (12 games) < < < < <
Uniform limit-cycle bounds in Hilbert's sixteenth problem HOT!	Banach's simple Lebesgue-spectrum problem	Rokhlin's multiple-mixing problem	Sinai's positive-entropy conjecture for the standard map
The near-boundary Birkhoff conjecture	The dimension formula for self-similar measures on the line	Permanence for weakly reversible reaction networks	Weak mixing of irrational triangular billiards
A $C^1$ counterexample to Shub’s entropy conjecture	Zero entropy does not guarantee a smooth positive-volume model	Arithmetic classification of Bernoulli convolutions	Pointwise multiple ergodic averages for mixing transformations

> > > > > Combinatorics (37 games) < < < < <
A counterexample to periodic tiling in dimension three HOT!	Borsuk's conjecture fails in dimension nine HOT!	Counterexamples to the Hadwiger and Colin de Verdière conjectures	The Euclidean plane cannot be colored with five colors PLAYABLE!
Erdős's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds	Superexponential van der Waerden numbers HOT!	Counterexamples to Sidorenko's conjecture and the forcing conjecture	Counterexamples to Ryser's covering and Gyárfás's tree-cover conjectures
Hindman's finite sums and products conjecture	Exact crossing numbers of complete and complete bipartite graphs	The higher-dimensional Erdős distinct-distances conjecture	Pinned distances and a power saving for planar unit distances
Combinatorial invariance of Kazhdan–Lusztig polynomials	The Shareshian–Wachs $e$-positivity conjecture	Sharp logarithmic exponents for off-diagonal Ramsey numbers HOT!	The hypercube Ramsey conjecture
Classification of finite Euclidean Ramsey configurations	Seymour's second-neighborhood conjecture HOT!	Deterministic construction of strong thin spanning trees	Talagrand's conjectures and graph decompositions at expectation thresholds
The second Kahn–Kalai conjecture	Bounded-degree coboundary expanders in every dimension	Deterministic nonbipartite Ramanujan graphs in every fixed degree	The circulant Hadamard conjecture HOT!
Barnette's Hamiltonian-cycle conjecture PLAYABLE!	The Erdős–Gallai cycle-decomposition conjecture	Power savings for polynomial-difference-free sets	Power savings for planar halving lines and $k$-sets
Coloring and independence in graphs with forbidden subgraphs	Counterexamples to infinite matroid intersection and packing/covering	The Friedgut–Kalai graph and hypergraph threshold conjectures	Snaky in 21 Maker moves PLAYABLE!
The sharp constant in random triangle removal	Exact cycle–clique Ramsey numbers	Polynomial removal fails for ordered binary matrices	A power improvement in the Heilbronn triangle problem
A counterexample to the Gopalan–Servedio conjecture

> > > > > Algebra (18 games) < < < < <
Serre's intersection-multiplicity conjecture	Lech's multiplicity conjecture	A counterexample to the small Cohen–Macaulay module conjecture	A counterexample to Kaplansky's zero-divisor conjecture HOT!
Nonsofic groups and group-ring counterexamples	A counterexample to the little finitistic-dimension conjecture	Counterexamples to conjectures of Auslander–Reiten, Tachikawa, and Nakayama	Eisenbud–Green–Harris and lex-plus-powers in characteristic zero
A counterexample to Kurosh's division-ring problem	The blockwise Alperin weight conjecture	Donovan's conjecture over fields and discrete valuation rings	Tensor saturation for even spin groups
Saxl's conjecture and universal tensor squares	Finite lattice representation: counterexamples and undecidability	The $\ell^1$-Bass and complex Bass trace conjectures	The finite Benson–Etingof–Ostrik conjecture
Integral counterexamples to Gersten’s conjecture	Foulkes’ conjecture for sixth powers and quadratic stabilization

> > > > > Probability and statistical mechanics (29 games) < < < < <
Geometry, diffusion, and spectra of random planar maps	No bigeodesics and smooth limit shapes in planar first-passage percolation	No infinite critical clusters on quasi-transitive graphs	The Benjamini–Schramm nonuniqueness conjecture
Massive continuum limits and exact mass asymptotics for planar $O(n)$ models	Critical logarithmic corrections and BKT scaling for the planar XY model	The low-temperature Sherrington–Kirkpatrick fluctuation law	Conformal universality for weakly interacting and disordered Ising models
GOE universality for random regular graphs with weak disorder	Directional zero–one laws and ballisticity in random environments	The Mézard–Parisi formula for diluted spin glasses	Perceptron free energies and microscopic jamming
Conformal limits of square-lattice random-cluster interfaces	Critical and near-critical universality for Voronoi percolation	Gaussian free field limits for the balanced six-vertex model	The double-dimer $\mathrm{CLE}_4$ conjecture in the half-plane
The dynamical phase transition in the Sherrington–Kirkpatrick model	Continuum phase transitions for radial pair potentials	Sharp three- and four-state reconstruction thresholds	Exact Hausdorff measure for SLE
The free uniform spanning forest is a factor of IID	Gaussian fields and SLE interfaces for Lipschitz heights	The joint critical Ashkin–Teller scaling limit	All-temperature pressure for orthogonally invariant Ising spin glasses
Random-SAT thresholds, sharp variance and computability	The factor-of-IID threshold for free Ising states on trees	The three-quarter diameter exponent for honeycomb walks	Optimal logarithmic mixing of the Thorp shuffle
Sharp singularity rates for symmetric sign matrices

> > > > > Mathematical logic (6 games) < < < < <
Shelah's eventual categoricity conjecture	Rigidity of the Turing degrees HOT!	Single-fold Diophantine representations	Choiceless counting does not capture polynomial time
The partition principle does not imply choice HOT!	The $\beta$-Barendregt–Geuvers–Klop conjecture

> > > > > Group theory (14 games) < < < < <
Cannon's conjecture HOT!	An infinite finitely presented residually finite $2$-group	Thompson's group $F$ is nonamenable HOT!	A finitely generated counterexample to Eilenberg–Ganea
The Boone–Higman conjecture and higher finiteness	Amenability, unitarizability, and Ulam stability	A non-residually-finite torsion-free hyperbolic group	An infinite finitely presented simple amenable group
Classifying spaces and geometric obstructions for Artin groups	Quasi-isometric rigidity of virtually polycyclic groups	Howie's conjecture on equations over groups	A hyperbolic group with no geometric CAT(0) action
Gersten's conjecture for one-relator groups	A group without fixed price

> > > > > Mathematical physics (25 games) < < < < <
Spacetime Penrose inequalities and rigidity	Localization and delocalization in the Anderson model	Sharp one-dimensional Lieb–Thirring inequalities	The ionization and generalized ionization conjectures
Strong cosmic censorship near two-ended Kerr data HOT!	The two-dimensional gapped area law	Exactly three mutually unbiased bases in dimension six HOT!	Positive-temperature Bose–Einstein condensation and quantum depletion
The spin-one Haldane gap	The Laughlin gap and stability under scalar disorder	Threshold and positive-energy bound states of the BFSS model	Bloch's law and spontaneous ferromagnetic order
Entanglement without secret key and the PPT-square conjecture	The entropy photon-number inequality	Moore's parity conjecture for $\mathrm{QAC}^0$	QMA-hardness of continuum Coulomb energy
The classical capacity of generalized amplitude damping	Threshold repetition for entangled games	Failure of Kohn–Sham ensemble representation	Exact quantum factoring over a fixed finite gate set HOT!
Strong locality for strongly rational unitary vertex operator algebras	QAOA optimality for the SK model	Scale-to-conformal enhancement in four-dimensional QFT	Polynomial-time, constant-error unitary synthesis from a Boolean oracle
The optimal randomized–quantum query exponent

> > > > > Operator algebras (19 games) < < < < <
Counterexamples to Baum–Connes and Kadison–Kaplansky	Arithmetic rigidity of lattice von Neumann algebras	All nonabelian free group factors are isomorphic HOT!	Kadison's similarity conjecture
The strong Kadison–Kastler conjecture	Connes' bicentralizer conjecture and relative bicentralizers	Toms–Winter and equivariant Jiang–Su stability	A counterexample to Kirchberg's norm-ultrapower embedding problem
A counterexample to the hyperinvariant-subspace problem HOT!	A counterexample to Kaplansky's quasitrace conjecture	The Kadison–Ringrose cohomology conjecture	The generator problem for finite factors
A ZFC counterexample to Naimark's problem	A counterexample to Voiculescu’s free-entropy equality conjecture	The Kirchberg–Rørdam character criterion	The Popa–Vaes quadratic strong-operator paving conjecture
Classification by trace cones after Razak–Jacelon stabilization	The Phillips–Toms formula for minimal integer actions	From ordinary to strong pure infiniteness

> > > > > Topology (18 games) < < < < <
The Hilbert–Smith conjecture in every dimension HOT!	Counterexamples to disk embedding and Wall's manifold conjecture	The purely cosmetic surgery conjecture for knots in $S^3$	Failure of rational injectivity for maximal coarse assembly
Smith–Toda complexes at every height with varying primes	The Kervaire invariant problem at the prime three HOT!	Quillen's conjecture in rational homology	Chai's invariant-ideal conjecture
The Grothendieck homotopy hypothesis HOT!	Finite generation for the $K(n)$-local sphere	The chromatic Smith fixed-point problem for finite $p$-groups	The four-dimensional Singer conjecture
Curtis’s conjecture	Thomason model structures in every strict higher dimension	Chromatic splitting: counterexamples and filtrations	Counterexamples to the Hahn–Wilson conjecture at height two
A four-dimensional counterexample to Borel rigidity	A counterexample to Wall's finite D(2) conjecture

> > > > > Functional analysis (11 games) < < < < <
Tingley's sphere-isometry problem	Relative independence of the separable quotient problem	Lipschitz equivalence without linear isomorphism	The complete Crouzeix conjecture
The cotype–cotype conjecture under the approximation property	Markov type characterizes superreflexivity	Fixed points of nonexpansive maps in reflexive Banach spaces	A counterexample to Pietsch's metric-entropy duality conjecture
A negative answer to Kalton's Lipschitz-free approximation question	Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming	Metric Markov cotype two for $\ell_1$

> > > > > Differential geometry (29 games) < < < < <
Smooth isometric immersions of surfaces into $\mathbb R^4$	A smooth surface metric with no local immersion in $\mathbb R^3$	Gromov's integral scalar-curvature inequality	Spectral scalar curvature and codimension-two width
Cartan–Hadamard isoperimetry and CAT$(0)$ fillings	Yau's uniformization conjecture	Katok's entropy rigidity conjecture	A counterexample to the nearby Lagrangian conjecture
Donaldson's hypersymplectic deformation conjecture	Donaldson's tamed-to-compatible conjecture	Sharp symplectic ball-packing criteria in higher dimensions	The metric Blaschke conjecture
Infinitely many closed geodesics on spheres and three-manifolds	Sharp singular-set bounds for stationary integral varifolds	Counterexamples to strong forms of Arnold's fixed-point conjecture	Nonnegatively curved Einstein four-manifolds and a topological gap
The Solomon–Yau least-volume conjecture	Yau's nodal conjecture: surfaces and counterexamples	Scalar curvature detects four-dimensional Ricci-flow singularities	Finite-time singularity of Calabi flow
The sharp dimension threshold for affine Bernstein rigidity	Isoperimetric regions in the cubic three-torus	Unique tangent flows at the first surface singularity	Gigli's characterization of Alexandrov curvature
Bi-Lipschitz coordinates at every regular RCD point	A three-manifold with no conjugate points and no nonpositively curved metric	Negative Kähler curvature without bounded holomorphic coordinates	Villani’s convexity conjecture and regular optimal transport
Counterexamples to Yau's harmonic dimension bound

> > > > > Partial differential equations (16 games) < < < < <
Global smoothness for relativistic Vlasov–Maxwell HOT!	Boltzmann nonuniqueness with exact local conservation	Kinetic limits and fluctuations over the Boltzmann lifespan	Joint metric and connection recovery from one boundary patch
The planar Mumford–Shah regularity conjecture	The critical dimension for the one-phase Bernoulli problem	The three-dimensional Ball–Evans approximation problem	The hot spots conjecture for simply connected planar domains PLAYABLE!
The subcritical Lane–Emden and Hénon–Lane–Emden conjectures	Stable blowup for a defocusing Schrödinger equation	Global uniqueness in smooth isotropic elasticity	Nonattainment in three-marginal Coulomb transport
Sharp one-third stability of optimal transport maps	De Giorgi's conjecture in dimension eight HOT!	Universal computation in forced Navier–Stokes flows HOT!	Interior $C^{1,\alpha}$ regularity for infinity-harmonic functions

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Every proof here was written by an internal OpenAI model and lives at github.com/openai/math.
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