A single black cell grows into a forty-year math mystery

Rule 30: one black cell

Start with one black square and apply Rule 30, a three-cell rule you could run with a pencil, and you get a pattern whose left side settles into stripes while its right side never settles at all. Stephen Wolfram's 2019 Rule 30 Prizes offer $30,000 for three questions about the centre column, and seven years on all three remain open. The page explores where order ends at the edges, a 56-step wheel that appears when the centre is forced to repeat, and why a finite start can only mimic any chosen sequence for about one step per square.