Q: 1, 1 1, 2 1, 1 2 1 1, ...

What is the pattern and asymptotics of this series?

A: Each line is derived from the last by the transformation (for example)

... z z z x x y y y ... -> ... 3 z 2 x 3 y ...

John Horton Conway analyzed this in "The Weird and Wonderful Chemistry of Audioactive Decay" (T M Cover & B Gopinath (eds) OPEN PROBLEMS IN COMMUNICATION AND COMPUTATION, Springer-Verlag (1987)). You can also find his most complete FRACTRAN paper in this collection.

First, he points out that under this sequence, you frequently get adjacent subsequences XY which cannot influence each other in any future derivation of the sequence rule. The smallest such are called "atoms" or "elements". As Conway claims to have proved, there are 92 atoms which show up eventually in every sequence, no matter what the starting value (besides <> and <22>), and always in the same non-zero limiting proportions.

Conway named them after some other list of 92 atoms. As a puzzle, see if you can recreate the list from the following, in decreasing atomic number:

U Pa Th Ac Ra Fr Rn Ho.AT Po Bi Pm.PB Tl Hg Au Pt Ir Os Re Ge.Ca.W Ta HF.Pa.H.Ca.W Lu Yb Tm ER.Ca.Co HO.Pm Dy Tb Ho.GD EU.Ca.Co Sm PM.Ca.Zn Nd Pr Ce LA.H.Ca.Co Ba Cs Xe I Ho.TE Eu.Ca.SB Pm.SN In Cd Ag Pd Rh Ho.RU Eu.Ca.TC Mo Nb Er.ZR Y.H.Ca.Tc SR.U Rb Kr Br Se As GE.Na Ho.GA Eu.Ca.Ac.H.Ca.ZN Cu Ni Zn.CO Fe Mn CR.Si V Ti Sc Ho.Pa.H.CA.Co K Ar Cl S P Ho.SI Al Mg Pm.NA Ne F O N C B Be Ge.Ca.LI He Hf.Pa.H.Ca.Li

Uranium is 3, Protactinium is 13, etc. Rn => Ho.AT means the following: Radon forms a string that consists of two atoms, Holmium on the left, and Astatine on the right. I capitalize the symbol for At to remind you that Astatine, and not Holmium, is one less than Radon in atomic number. As a check, against you or me making a mistake, Hf is 111xx, Nd is 111xxx, In and Ni are 111xxxxx, K is 111x, and H is 22.

Next see if you can at least prove that any atom other than Hydrogen, eventually (and always thereafter) forms strings containing all 92 atoms.

The grand Conway theorem here is that every string eventually forms (within a universal time limit) strings containing all the 92 atoms in certain specific non-zero limiting proportions, and that digits N greater than 3 are eventually restricted to one of two atomic patterns (ie, abc...N and def...N for some {1,2,3} sequences abc... and def...), which Conway calls isotopes of Np and Pu. (For N=2, these are He and Li), and that these transuranic atoms have a zero limiting proportion.

The longest lived exotic element is Methuselum (2233322211N) which takes about 25 applications to reduce to the periodic table.

- Matthew P Wiener (weemba@libra.wistar.upenn.edu)

Conway gives many results on the ultimate behavior of strings under this transformation: for example, taking the sequence derived from 1 (or any other string except 2 2), the limit of the ratio of length of the (n+1)th term to the length of the nth term as n->infinity is a fixed constant, namely

1.30357726903429639125709911215255189073070250465940...

This number is from Ilan Vardi, "Computational Recreations in Mathematica", Addison Wesley 1991, page 13.

Another sequence that is related but not nearly as interesting is:

1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314,

and 21322314 generates itself, so we have a cycle.