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.
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.
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.
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.