Multiply an irrational number by all powers of 2 and 3 simultaneously: must the fractional parts be dense between 0 and 1? Two commuting actions that should jointly mix — and stubbornly refuse to prove it.
The Problems
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.
Difficulty 4/5 — 33 shown. Clear
Fuglede's Conjecture
historic
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.
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.
The Willmore Conjecture
historic
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.
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.
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.
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.
Hedetniemi's Conjecture
historic
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.
Numbers \(2^p\) − 1 that are prime power the hunt for ever-larger known primes — yet we cannot prove the supply is infinite.
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.
Beal's Conjecture
open
\(A^x\) + \(B^y\) = \(C^z\) 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.
Odd Perfect Numbers
open
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.
Hilbert's Tenth Problem
historic
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.
The Erdős Discrepancy Problem
historic
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.
Keller's Conjecture
historic
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.
The Fifth Busy Beaver Number BB(5)
historic
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.
The Moving Sofa Problem
historic
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.
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.
The Cap Set Problem
open
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.
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.
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.
The Kepler Conjecture
historic
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.
The Poincaré Conjecture
historic
Is a simply-connected closed 3-manifold necessarily a sphere? Ninety-nine years, one recluse, and the only Millennium Prize ever claimed.
Multicolour Triangle Ramsey Numbers (Erdős #183)
AI-resolved
How many colours force a monochromatic triangle? Erdős's fifty-year-old lower-bound challenge, resolved superexponentially by Astra.
Feige's Anticoncentration Conjecture
AI-resolved
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.
The Cycle Double Cover Conjecture
AI-resolved
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.
Erdős's Unit Distance Problem
AI-resolved
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.