Start here

A rule you could run with a pencil

Take a strip of squares, each black or white. To make the next strip, look at each square together with its two neighbours, and colour the square below by one fixed rule. Rule 30 is one such rule, one of 256, and it is the one that makes the simplest possible beginning look like noise.

The rule

Eight possible neighbourhoods, eight answers. In words: the new square is black when the left neighbour differs from "the middle or the right". Written as a formula, new = left XOR (centre OR right).

One more way to read it. Drop the "or the right" and the rule becomes left XOR centre, which draws the Sierpinski triangle. Rule 30 is that triangle rule plus a carry that fires whenever the right neighbour is black twice running, and that carry is where all the disorder comes from. The carry, drawn and checked.

The pyramid

Nothing random went in. The rule is fixed, the start is one square, and every row is determined by the one above it. Yet the right side has passed every statistical test for randomness that has been thrown at it, and Rule 30 was for years the random number generator inside Mathematica.

The three questions, as a coin

Read the centre column as a coin: black is heads, white is tails. Stephen Wolfram's prize, announced in 2019, offers ten thousand dollars for a proof of each of three statements about that coin.

1. It never settles. Does the sequence ever fall into a repeating pattern that goes on for ever? The claim is that it does not. This is the question we worked on.
2. It is fair. In the long run, are heads and tails equally common? The claim is that they are.
3. There is no shortcut. Does finding the n-th flip take about n steps of work, however clever you are?

The coin is not random, so each answer is simply true or false. Only a proof settles it, and no amount of computation can: ten million flips without a pattern says nothing about the ten million and first.

Sources. Wolfram's 2019 announcement at rule30prize.org; the coin wording is from STATE-OF-THE-PROOF.md section 2 and RULE30-PRIZE.md, "The three questions, as a coin".

Why it is hard

The first question, taken one period at a time, is the one this record attacks. A pattern of period one would mean the centre column eventually stays one colour for ever. That case was closed in September 2026, before we began: Condrey proved that no column of a finite starting pattern can become constant. The next case is period two, a centre that reads black, white, black, white for ever, and it is open. Everything on this site about where we stand is about that case.

Here is the obstacle in one sentence. The rule is so simple that running it forwards tells you nothing new, and so mixing that every known tool for proving things about "almost all" starting patterns says nothing about the one starting pattern the prize is about. What a proof needs is a reason, true for the single pattern that starts from one square, why keeping a repeating centre for ever costs more than the pattern can pay. Where that reason stands. To see the obstacle live, hold any repeating word down the centre and watch the column beside it refuse to repeat.

Sources. Condrey, arXiv:2609.09431 (2026), and Jen (1990), as cited in PRIOR-ART.md; the one-sentence form is PERIOD-TWO.md section 5.

More to explore

The full worked story of Rule 30, with ten interactive visuals and sound, written for a reader with no background, is part of this site: the Wolfram Rule 30 page.