Wolfram · Rule 30 · $10,000 a question

Rule 30 and Wolfram's three questions

Rule 30 is a rule you could run with a pencil. A row of black and white squares makes the next row: each new square looks at the three above it. Start from a single black square and a pyramid grows. Its left side is orderly, but its right side looks like static. Down the middle runs a column that behaves like a fair coin. In 2019 Stephen Wolfram offered $10,000 for a proof of each of three statements about that column. All three are open. This project works on the first, one period at a time.

A more detailed walkthrough of the Rule 30 problem can be found here: the whole problem from the first row to where Question 1 stands, with ten interactive pictures and sound. No background needed.

Seen

Rule 30 from one black cell, growing row by row. The tinted column down the middle is the one the prize is about.

The first 120 cells of that column, read left to right: black for heads, white for tails.

Stated

The three prize questions (Wolfram, 2019)

1. It never settles. The centre column never becomes periodic: it never falls into a repeating pattern for ever. 2. It is fair. In the long run, black and white are equally common. 3. There is no shortcut. Computing the n-th cell of the column takes about n steps of work, however clever the method.

Where Question 1 stands

Period one is settled. Condrey proved in September 2026 that no column of a finite starting pattern can become constant. We proved it again by hand, and it is machine-checked in Lean. Period two is open. It asks whether the centre can read black, white, black, white for ever. Nearly all our work went here. We proved that whole families of would-be counterexamples cannot exist. We computed how far any counterexample must sit from the centre: no left edge within 248 cells, whatever the right half. And we certified every word of length up to 44 that the next column can never produce.

The one missing statement

Let R(d) be the longest white run that the forced left side can show at depth d, over all real traces beside the alternating centre. If R(d) is finite at every depth, period two is impossible for every finite starting pattern. The exact values are known at every depth to 120 and at six depths beyond. Proving finiteness at every depth is the open part. It needs a reason why keeping a repeating centre for ever costs more information than a finite pattern has, and no field of mathematics has a method for that yet.

Sources. Wolfram's announcement at rule30prize.org; STATE-OF-THE-PROOF.md; PERIOD-TWO.md sections 2, 5 and 6; RULE30-PRIZE.md sections 8.56 and 8.76 (the 248-cell search; period one again); Condrey, arXiv:2609.09431.

Checked, here, now

Question 2 says the coin is fair. Pick a length and your browser runs Rule 30 from one black cell, reads the centre column, and counts.

Steps

Fairness so far is not a proof: Question 2 is about the limit, and no finite count can settle it. Over 2¹⁷ steps the record measures a density of 0.49947, with block statistics like a fair coin's (RULE30-PRIZE.md section 8.34).

Where to go next

Where we stand: what is proved, what is measured, which routes are closed, and why.
Live pictures: the forced left half, a periodic centre making its own chaos, the record route, the ring of six and the carry of Rule 30.
For mathematicians: the open statements in symbols.
How the three problems connect: Rule 30's question has the shape of Collatz and Mahler's 3/2 problem.