Ninety-Two Elements,
Hiding in a Number

Here is the whole game. Take a string of digits. Say what you see, out loud, in runs: 1 is one 1, so it becomes 11. That is two 1s, so 21. That is one 2, one 1, so 1211. Then one 1, one 2, two 1s: 111221. There are no sums, no primes, no magic. You are only ever describing the line above.

John Conway took up this children's puzzle in late 1983 and reported finding its elements in spring 1984. His article appeared in 1986. Play first, then inspect the live computations, offline derivation, and cited history.

Watch the length. It roughly multiplies by the same amount at every step. Change the seed to anything you like, 3, 1112, your phone number, and the multiplier settles to the identical value. Conway proved it: for every starting string except two degenerate cases, the ratio of successive lengths converges to one number.

It is called Conway's constant, written λ, and it is about 1.3035772690. Below, every seed you throw at it (the bright violet line is yours, the faint lines are other starts) bends toward the same gold level.

Here is Conway's stranger discovery. Take a long term and look for places where the string splits into two halves that, from then on, never influence each other again no matter how many times you say-what-you-see. We call such a boundary a wall. Conway called the separation a splitting, and a string with no nontrivial splitting an atom, or element.

Below, your string is cut at certified walls into colored chunks. The bounded test can leave an unconfirmed boundary unsplit, so a chunk may contain more than one atom. Press decay and each chunk independently becomes the next generation. The analogy is radioactive decay: elements breaking into other elements. Here every click advances the whole compound by one generation. Conway named it audioactive decay, because you make it happen by reading aloud.

Chase the decay of any ordinary seed long enough and it settles into a soup of atoms drawn from a fixed, finite set. Conway proved the set has exactly 92 common elements, and, with a wink, named them after the first 92 chemical elements, Hydrogen through Uranium. This is his Cosmological Theorem: every look-and-say sequence, whatever its seed, decays within a bounded number of steps into a compound of these 92 (plus the transuranics if you used a big digit).

Here they are, derived offline from scratch and shaded by how common each is in a long string. Click any tile.

Everything above is a claim. Here is why you can trust it: three approaches to the same value. The count and matrix share the look-and-say operator and seed 1; the checks also share reference digits. The matrix and polynomial values agree to twelve decimal places; the counted ratios approach them. The full script is research/conway-audioactive/verify.mjs in this project's repository, run offline, 22 of 22 checks passing.

And the periodic table itself is checked, not asserted. The 92 elements on this page were not copied from a reference; the verifier finds the walls of the sequence from first principles, splits it into atoms, takes the closure under decay, and counts 92. Their abundances (the eigenvector of Road 2) reproduce Conway's published values at every named anchor: Hydrogen 91790.383, Protactinium 9883.599, Helium 3237.297, Uranium 102.563, down to Arsenic 27.246, each to one part in ten thousand. These checks corroborate the derived table and the tested boundaries; they do not certify the splitter on every input.

The Cosmological Theorem is genuinely a theorem, but its history is a small scandal. Conway and Richard Parker gave a first proof; Mike Guy gave a stronger one (that 24 steps always suffice, and that 24 is best possible). Both proofs were then lost. The result sat re-provable but unreproved until 1997, when Shalosh B. Ekhad and Doron Zeilberger published a computer proof, "Proof of Conway's Lost Cosmological Theorem"; R. A. Litherland later gave a second, independent machine proof with tighter bounds.

One more honesty note on the polynomial. Conway's degree-71 polynomial was printed correctly in his 1986 Eureka article but misprinted in the widely cited 1987 reprint. The coefficients this page uses are OEIS A137275, and the verifier confirms both that they are monic of degree 71 and that λ is their root; do not trust the 1987 book's version.

A constant like π shows up in circles because circles are round. Conway's constant shows up in a game with no geometry, no arithmetic, nothing but the act of describing the previous line. It is an algebraic number of degree 71, meaning it satisfies a polynomial of degree 71 and no smaller one, so it can be pinned down exactly yet has no tidy closed form, a cousin of the irrationals in Incommensurable. Each term of this sequence describes the preceding term, and out of that description falls a fixed number and a closed chemistry of 92 common element types. On the linked Shannon page, prediction and counting instead estimate a band for the entropy of English. Conway's own summary, in the paper's title, is hard to improve on: the weird and wonderful chemistry of audioactive decay.