Iterated absolute differences of the primes always start with 1. Verified to 10^13 and beyond, proved nowhere — perhaps the most elementary open problem about the primes.
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 2/5 — 6 shown. Clear
Chomp is a first-player win, but nobody knows the winning first move in general. Strategy-stealing proves existence; explicit strategy unknown.
Morley's Trisector Theorem
historic
In ANY triangle, the three adjacent angle trisectors meet pairwise at the vertices of an equilateral triangle. Discovered around 1874, published 1899 — pure Euclidean magic with no construction possible by compass alone.
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.
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.
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.