Puzzles
A bored mathematician doodled a spiral and stumbled onto an unsolved mystery
Write the positive integers in a square spiral starting from 1 at the centre, then circle only the prime numbers. Do the circled primes scatter randomly across the grid, or is there a pattern?
Reveal the answer
They cluster visibly along certain diagonal lines, far more than chance would suggest. Stanislaw Ulam noticed this in 1963 while doodling during a tedious conference talk. The clustering is linked to the unproven Hardy–Littlewood 'Conjecture F' about how often certain quadratic formulas produce primes — nobody has ever proven why the diagonals line up the way they do.