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.
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.
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.
The Weil Conjectures
open
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.
Moser's Worm Problem
open
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.
The Tate Conjecture
open
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.
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.
Sendov's Conjecture
open
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.
Cramér's Conjecture
open
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.
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.
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.
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.
Ramsey Number R(5,5)
open
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.
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.
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?
The abc Conjecture
open
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.
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.
The Hodge Conjecture
open
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?
P versus NP
open
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.