Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Euler proved ζ(2) = π²/6 in 1735. Whether ζ(3) is rational stayed open for 243 years — until Roger Apéry announced a miraculous recurrence-driven proof at 64 years old.
Should a set admit a Fourier basis exactly when it tiles by translations? Spectacularly false in dimension ≥ 3 — Terence Tao's counterexample — while the low-dimensional wreckage keeps generating theorems.
Any dense subset of the n-dimensional tic-tac-toe board contains a whole line. Proven by the FIRST-ever Polymath project — dozens of mathematicians thinking in one comment thread.
An angel jumps k squares per move eating tiles; a devil burns one square forever. Can the angel escape forever? Conway offered $100 — four independent proofs arrived within months in 2006.
Given a group presentation and a word, decide if the word equals the identity. Dehn asked for algorithms; logic answered that none can exist — for some groups.
Are finitely generated groups of bounded exponent necessarily finite? No — wildly no — yet their RESTRICTED cousin said yes so profoundly it earned a Fields Medal.
Among all closed surfaces in 3-space, does the Clifford torus minimise bending energy (∫ H²)? Proven by min-max theory fifty years later, in work that revived geometric measure theory.
Can pure states extend uniquely to a bigger algebra? A quantum-measurement question from 1959, resolved in 2013 by undergraduates-will-understand polynomial inequalities — with ripple effects across engineering.
Counting solutions of equations over finite fields should follow deep topology: zeta functions satisfy RH-analogues because varieties behave like classical manifolds. Deligne's 1974 proof reshaped everything.
How often can the SAME integer appear in Pascal's triangle? Singmaster guessed a uniform bound; the number 3003 appears six times and nobody can rule out seven.
Does a box exist with integer edges, integer face diagonals AND integer space diagonal? Three centuries of searching; not one example, no proof of impossibility.
A precise promise problem about labelling cycles of pairwise consistency checks that is believed (was believed?) NP-hard — and became the load-bearing assumption for half of optimal-inapproximability theory.
How many sets of size w force a sunflower (petals meeting pairwise in the same core)? Should be c^w; sixty-four years of effort moved the base from w down to O(log w) — never to a constant.
Label any tree's vertices 1..n so that edge-differences are all distinct. Sixty years of near-misses culminated in November 2025: every large tree gets ALMOST there.
Smallest convex blanket covering EVERY set of diameter 1? Pál's 1920 regular hexagon has been shrinking for a century — most recently in 2024.
Find the smallest-area blanket that can cover a unit-length curve no matter how it bends. Sister puzzle to our solved moving sofa — and still wide open.
γ = lim(1+1/2+...+1/n − log n) ≈ 0.5772…: we cannot even prove it is IRRATIONAL, let alone transcendental. Mathematics' most embarrassing constant.
Describe ALL abelian extensions of a number field explicitly, the way roots of unity generate them for Q. Kronecker called it his 'dream of youth'; it built class field theory and still dreams on.
Algebraic subvarieties should be visible in étale cohomology: which cohomology classes come from actual cycles? The arithmetic mirror of the Hodge conjecture, and nearly as stubborn.
Primes are equidistributed in arithmetic progressions on average up to modulus q^{1−ε} for EVERY ε — twice the reach of the Bombieri–Vinogradov theorem. Half the modern theory of primes leans on this doubling.
The Liouville function λ(n) should look random: correlations \sum_{n≤x} λ(n+h_1)...λ(n+h_k) should be o(x) for any fixed shifts. Two-point progress exists; three points and beyond resist.
Bound the number of limit cycles of a polynomial vector field in terms of its degree. Over 120 years on, not even the quadratic case has a complete answer.
Can ONE single shape tile the plane, but only aperiodically? Yes — the 13-sided "hat", found in 2023 by a hobbyist mathematician and his collaborators, ending a sixty-year quest.
Is the colouring cost of a tensor product of graphs always the cheaper factor\'s? Fifty-three years of belief ended with Yaroslav Shitov\'s three-page demolition.
Must every bounded set in n-dimensional space split into n+1 pieces of smaller diameter? True in low dimensions — spectacularly false in high ones.
The E₈ and Leech lattices are THE tightest packings of balls in their dimensions — proven by Maryna Viazovska\'s magic modular-form functions in a fortnight that stunned mathematics.
How wildly does the Riemann zeta function grow on the critical line? Lindelöf said barely at all — a consequence of RH that may be provable even without it.
Every root of a polynomial with all roots in the unit disk should lie within distance 1 of SOME critical point. Gauss-Lucas says critical points live in the disk; Sendov asks for the finer choreography.
Can a graph drawn so that every edge meets every other edge exactly once have more edges than vertices? Conway bet $1,000 nobody finds one.
Numbers 2^p − 1 that are prime power the hunt for ever-larger known primes — yet we cannot prove the supply is infinite.
How far apart can consecutive primes be? Cramér's probabilistic model says never much more than (log p)² — a prediction we cannot approach unconditionally.
How few distinct distances must n plane points determine? Guth and Katz nearly closed it in 2010 — leaving only a logarithm standing between us and the answer.
The last survivor: after the three-variable case fell to a 216-character counterexample in 2026, the ORIGINAL two-variable Jacobian question remains open.
Can you always rebuild a graph from its deck of vertex-deleted cards? Ulam-style determinism for combinatorial structure — open for over eighty years.
Does every graph needing k colours contain k pairwise vertex-disjoint connected subgraphs mutually joined by edges? One conjecture that would explain ALL of colouring.
A^x + B^y = C^z with all exponents greater than 2 forces a common factor among the bases. Fermat's Last Theorem's living descendant, with a million-dollar bounty.
Perfect numbers equal the sum of their proper divisors. Euclid knew even ones come from Mersenne primes; whether an odd perfect number exists has outlived twenty-three centuries.
How algebraically independent can exponentials of numbers be? The master key to transcendence — prove it and Lindemann-Weierstrass, four-exponentials and much more fall out at once.
Why is there no formula in radicals for fifth-degree equations? Abel proved none exists; Galois explained exactly why — and invented group theory doing it.
Trisect the angle, double the cube, square the circle — compass and straightedge alone. Two millennia of failure resolved by nineteenth-century algebra: all three are impossible, and the proofs founded modern field theory.
Give an algorithm deciding whether a polynomial has integer roots. Impossible — and proving impossibility wove Diophantine equations, Fibonacci numbers and computability into one of the 20th century\'s great joint theorems.
Is there a cardinality strictly between the integers and the reals? The first Hilbert problem dissolved into independence: yes in Gödel\'s universe, no in Cohen\'s — and both are legitimate.
Must every infinite ±1 sequence contain arbitrarily long homogeneous arithmetic progressions? Yes — Terence Tao's six-page proof ended an 83-year-old favourite, catalysed by a Polymath experiment.
S(5) = 160: the integers 1..160 can be 5-coloured with no monochromatic solution to x + y = z, but 1..161 cannot. A century-old Ramsey constant pinned by SAT.
In any tiling of space by unit cubes, must two cubes share a whole face? True up to dimension 6, false from 8 — dimension 7 held out until SAT solvers plus Lean closed it in 2020.
Can the integers be two-coloured with no monochromatic Pythagorean triple? No — and proving it required a 200-terabyte certificate, the largest proof in history at publication.
Which convex pentagons tile the plane? Exactly fifteen families — the final word delivered by Michaël Rao\'s exhaustive computer-assisted elimination.
BB(5) = 47,176,870: the maximal steps a halting 5-state Turing machine can run. Determined by a decade-long collaborative quest, machine-checked in Coq.
What is the largest area of a rigid shape that can negotiate an L-shaped corridor of unit width? Fifty-eight years of sofa-shaping ended with Jineon Baek's 119-page proof.
Is the 4-sphere, up to diffeomorphism, the unique smooth closed simply-connected 4-manifold? Topology says yes; smooth structures say nobody knows — and dimension four is where smoothness goes feral.
Colour the edges of complete graphs red/blue: how large before a monochromatic K₅ is forced? For K₅ versus K₅ we know only 43 ≤ R(5,5) ≤ 48.
In any non-empty family of sets closed under union, must some element appear in at least half the sets? Perhaps the simplest open statement in extremal set theory.
What is the least exponent ω with n×n matrices multipliable in O(n^ω) operations? Fifty-six years of algebraic ingenuity brought ω from 3 down toward 2 — never reaching it.
How large can a subset of F_3^n be containing no three-term arithmetic progression? Bounds have collapsed twice — once by the polynomial method, once to DeepMind's FunSearch.
Can every fraction 4/n be written as a sum of exactly three unit fractions? Egyptian mathematics meets the distribution of prime factors.
Does every simple closed curve in the plane contain four points forming a square? Over a century old, proven true for vast classes of curves, false for none.
How many colours suffice so that no two points exactly one unit apart share a colour? Between five and seven since 2018 — when Aubrey de Grey shook a 70-year-old stalemate.
n runners start together on a circular track, each with a distinct constant speed. Must every runner, at some moment, be strictly farther than 1/n of the track from all the others?
a + b = c: how large can the radical of abc be relative to c? A statement about the deep tension between addition and multiplication — and home of mathematics' most controversial claimed proof.
Does every bounded linear operator on a Hilbert space send some non-trivial closed subspace to itself? Fifty years of counterexamples on wilder spaces, two unrefereed claims on Hilbert space itself — and still no verdict.
How small can a set be that contains a unit line segment pointing in every direction? Dimension three fell in 2025 in a proof hailed as a once-in-a-century achievement; dimension four and beyond remain open.
Every odd number greater than 5 is the sum of three primes. Two hundred seventy-one years from letter to theorem — the little sibling of our unsolved shelf's Goldbach entry.
Grocers stack oranges in hexagonal layers for good reason: Kepler said no arrangement beats face-centred cubic packing at 74.05%. Proof took 387 years and a computer.
Is a simply-connected closed 3-manifold necessarily a sphere? Ninety-nine years, one recluse, and the only Millennium Prize ever claimed.
Four colours suffice for any map. The first major theorem proved by a computer — and the first mathematical controversy about what a proof is.
No positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. Three hundred fifty-eight years, ten million pages of attempted proofs, one margin note too long to contain.
A divisibility question about shifted powers — the first Erdős problem genuinely resolved end-to-end by an LLM pipeline, certified by a second AI and checked by humans.
How many colours force a monochromatic triangle? Erdős's fifty-year-old lower-bound challenge, resolved superexponentially by Astra.
Are the hyperfinite II₁ factor's matrix approximations unique — must certain groups be visible in their von Neumann algebras? Disproved by OpenAI's Astra model, ending a fifty-year expectation.
Are all finitely generated groups approximable by finite ones? OpenAI's Astra model announces a construction proving not — resolving a central question of group theory.
Sums of small independent random variables resist concentrating too sharply. A decade-plus-old conjecture, initially proved by ChatGPT — and by two independent teams the same day.
Must a finite set of reals grow fast under addition or multiplication? Over the reals: no — a cascade begun by AI's unit-distance breakthrough. Over the integers the conjecture lives on.
Can every bridgeless graph be covered by circuits traversing each edge exactly twice? Yes — with at most eight loops. Settled by GPT-5.6 Sol after fifty years.
If a polynomial self-map of space has everywhere-invertible derivative, must it be globally invertible? False in three variables: a 216-character counterexample ended an 87-year hunt.
How many pairs of n plane points can sit exactly one unit apart? Erdős guessed his grid was unbeatable. An AI model proved him wrong — the first prominent open problem resolved autonomously by AI.
Give the quantum field theory behind the Standard Model a rigorous foundation, and explain why the gluon is massive despite being massless in the equations.
On algebraic varieties, the analytic topology and the algebraic geometry see the same cycles. Whether they truly align is one of geometry's deepest gaps.
An elliptic curve has infinitely many rational points precisely when its L-function vanishes at s = 1. Deep arithmetic meets deep analysis.
Do solutions of the equations that describe fluid motion always exist and stay smooth, or can they blow up? Mathematics has not caught up with the water in the glass.
There are infinitely many primes p such that p + 2 is also prime. Infinitely close, structurally — but unproven.
Every even integer greater than 2 is the sum of two primes. The oldest surviving unsolved problem in number theory.
Take any positive integer: halve it if even, else triple it and add one. Does the sequence always reach 1?
If a solution can be checked quickly, can it also be found quickly? The question that organises theoretical computer science.
All non-trivial zeros of the Riemann zeta function lie exactly on the critical line Re(s) = 1/2. The single most consequential open statement in pure mathematics.