Rule 30: one black cell

The page explains Rule 30, a one-dimensional cellular automaton defined by the eight-bit rule 00011110 applied to rows of black and white squares starting from a single black cell. That simple local rule produces a triangular pattern whose left side settles into regular diagonal stripes while the right side remains chaotic; the central vertical column, when read as a sequence of black/white, passes standard randomness tests and resists proof-based classification. Stephen Wolfram offered $30,000 in 2019 across three precise problems about that centre column - whether it is eventually periodic, whether black and white occur with equal frequency, and whether computing the nth entry requires at least n steps - and extensive computations (billions of cells) show apparent randomness but do not answer the questions.

Detailed structural findings narrow where order exists: every diagonal parallel to the right edge repeats with a period that is a power of two, producing a provable ruler-like sequence of white triangles, while many left-edge diagonals settle into eternal stripes and the boundary between ordered and disordered diagonals advances at about 0.244 squares per row (a measured random-walk-like drift). If the centre were forced into a simple repeat, the adjacent column falls into a 56-step cycle decomposable into 8- and 7-position clocks, and a repeating centre plus repeating neighbor would crystallize the entire left side. Any infinite leftward start can force the centre to output any sequence; finite starts empirically buy roughly one centre step per starting square. Several of these behaviors are proved, others are measurements, and the core centre-column problems remain unresolved.