Seven complete sequences for the Conway Look and Say elements

Introduction

Conway's audioactive Look and Say generating algorithm evolves to a string of digits made up of non-interacting elements. 92 of these elements are "natural" in the sense that they are created from any starting string (other than "22") and they dominate the string after sufficient evolution. This number 92 suggests the atomic elements up to Uranium from the periodic table, and the evolution can be compared with radioactive fission (a pun with Look and Say and radioactive leads to the name audioactive).

The 92 elements can be ordered in a sequence so that the audioactive evolution goes through each element in turn (while also producing by-products). Mario Hilgemeier has drawn a pictorial graph of the transition, as part of his writings in Fractal Horizons. He asked whether the sequence is unique, and if not, then how many sequences there are with this property.

The answer is that it is not unique; there are seven possible such sequences. This page states the seven sequences and uses two different proofs to show they are the only such possible sequences. Step 0 sets up the proof and provides some simple rules which simplify the search for solutions. Step 1 shows how the sequence can be broken down into nine part sequences which must be contained in any complete sequence of the elements. Step 2 lists these nine part sequences and their possible sources and outlets. Step 3 uses brute force to discover which sequences work and which not. Step 4 is an alternative to Step 3, using rules and deduction to the same effect. Step 5 lists the seven possible complete sequences of the 92 elements identified in Steps 3 and 4.


Step 0

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A complete sequence is a full ordering of the 92 elements where each successive element produces the next directly, and where there is no duplication.

Every element in a complete sequence must have a single source and a single outlet, apart from the first (which only needs an outlet) and the last (which only needs a source). If one element is the source of a second element, then the second is the outlet of the first. 1H is stable and has no other outlet, and so must be the last element.

39Y has two possibilities, 38Sr and 92U, and is the only possible source for each; a complete sequence must therefore start with either 38Sr or 92U and every other element must have a source.

Apart from 1H, part sequences cannot loop back or return to themselves, since to do so would lead to duplication before completion of the complete sequence.


Step 1

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Using Step 0, we can be certain that a complete sequence contains certain definite part sequences:

92U-73Ta

92U only goes to 91Pa (so 91Pa not from 73Ta, 21Sc, 2He)
91Pa only goes to 90Th
90Th only goes to 89Ac (so 89Ac not from 31Ga)
89Ac only goes to 88Ra
88Ra only goes to 87Fr
87Fr only goes to 86Rn
86Rn only source for 85At (so 86Rn not to 67Ho)
85At only goes to 84Po
84Po only goes to 83Bi
83Bi only source for 82Pb (so 83Bi not to 61Pm)
82Pb only goes to 81Tl
81Tl only goes to 80Hg
80Hg only goes to 79Au
79Au only goes to 78Pt
78Pt only goes to 77Ir
77Ir only goes to 76Os
76Os only goes to 75Re
74W only goes to 73Ta (so return from 73Ta to 74W would lead to duplication and is impossible)
75Re only other source for 74W (so 75Re not to 32Ge, 20Ca)

72Hf-68Er

72Hf only goes to 71Lu
71Lu only goes to 70Yb
70Yb only goes to 69Tm
41Nb only source for 40Zr (so 41Nb not to 68Er)
69Tm only other source for 68Er

67Ho-64Gd

67Ho must got to 66Dy
66Dy only goes to 65Tb
65Tb only source for 64Gd (so without impossible return from 65Tb to 67Ho)

63Eu-62Sm

63Eu only goes to 62Sm

61Pm-39Y

61Pm only goes to 60Nd
60Nd only goes to 59Pr
59Pr only goes to 58Ce
58Ce only source for 57La (so 58Ce not to 27Co, 20Ca, 1H)
57La only goes to 56Ba
56Ba only goes to 55Cs
55Cs only goes to 54Xe
54Xe only goes to 53I
53I only source for 52Te (so 53I not to 67Ho)
52Te only source for 51Sb (so 52Tb not to 63Eu, 20Ca)
51Sb only source for 50Sn (so without return from 51Sb to 61Pm)
50Sn only goes to 49In
49In only goes to 48Cd
47Ag only goes to 46Pd
46Pd only goes to 45Rh
45Rh only source for 44Ru (so 45Rh not to 67Ho)
40Zr only source for 39Y (so 40Zr not to 43Tc, 20Ca, 1H)
44Ru only other source for 43Tc (so 44Ru not to 63Eu, 20Ca)
43Tc only goes to 42Mo
42Mo only goes to 41Nb
41Nb only source for 40Zr (so 41Nb not to 68Er)

38Sr-31Ga

38Sr only goes to 37Rb
37Rb only goes to 36Kr
36Kr only goes to 35Br
35Br only goes to 34Se
34Se only goes to 33As
75Re only other source for 74W apart from impossible 73Ta (so 75Re not to 32Ge, 20Ca)
4Be only other source for 3Li apart from impossible 2He (so 4Be not to 32Ge, 20Ca)
33As only other source for 32Ge (so 33As not to 11Na)
32Ge only source for 31Ga (so 32Ge not to 67Ho)

30Zn-21Sc

30Zn only goes to 29Cu
29Cu only goes to 28Ni (so return from 28Ni to 30Zn would lead to duplication and is impossible)
28Ni only alternative is to go to 27Co (so 27Co not from 69Tm, 64Gd, 58Ce, 21Sc)
27Co only goes to 26Fe
26Fe only goes to 25Mn
25Mn only source for 24Cr (so 25Mn not to 14Si)
24Cr only goes to 23V
22Ti only goes to 21Sc

20Ca-2He

20Ca only goes to 19K
19K only goes to 18Ar
17Cl only goes to 16S
16S only goes to15P
25Mn only source for 24Cr (so 25Mn not to 14Si)
15P only other source for 14Si (so 15P not to 67Ho)
14Si only goes to 13Al
13Al only goes to 12Mg
33As only other source for 32Ge apart from impossible 75Re and 4Be (so 33As not to 11Na)
12Mg only other source for 11Na (so 12Mg not to 61Pm)
11Na only goes to 10Ne
10Ne only goes to 9F
9F only goes to 8O
8O only goes to 7N
7N only goes to 6C
6C only goes to 5B
5B only goes to 4Be
3Li only source for 2He (so 2He cannot return to 3Li)
4Be only other source for 3Li apart from impossible 2He (so 4Be not to 32Ge, 20Ca)

1H

1H only goes to 1H and is therefore stable


Step 2

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So the part sequences are:

92U-73Ta which could come from 61Pm-39Y and lead to 72Hf-68Er, 20Ca-2He or 1H

72Hf-68Er which could come from 92U-73Ta or 20Ca-2He, and lead to 67Ho-64Gd or 61Pm-39Y

67Ho-64Gd which could come from 72Hf-68Er or 30Zn-21Sc, and lead to 63Eu-62Sm or 20Ca-2He

63Eu-62Sm which could come from 67Ho-64Gd or 38Sr-31Ga, and lead to 61Pm-39Y, 30Zn-21Sc or 20Ca-2He

61Pm-39Y which could come from 72Hf-68Er or 63Eu-62Sm, and lead to 92U-73Ta or 38Sr-31Ga

38Sr-31Ga which could come from 61Pm-39Y and lead to 63Eu-62Sm, 30Zn-21Sc, 20Ca-2He or 1H

30Zn-21Sc which could come from 63Eu-62Sm or 38Sr-31Ga and lead to 67Ho-64Gd, 20Ca-2He or 1H

20Ca-2He which could come from 92U-73Ta, 67Ho-64Gd, 63Eu-62Sm, 38Sr-31Ga or 30Zn-21Sc, and lead to 72Hf-68Er or 1H (but could not come from or lead to 20Ca-2He)

1H which could come from 92U-73Ta, 38Sr-31Ga, 30Zn-21Sc, 20Ca-2He, or 1H, and lead to 1H


Step 3

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Brute force (remembering from Step 0 that we must start from 92U or 38Sr) will reveal which combinations from Step 2 lead to duplications before completion, which ones arrive at 1H before visiting all of the other elements, and which are the seven complete sequences.


Step 4

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We could alternatively discover rules in addition to Steps 0, 1 and 2 which constraining the complete sequences, and thus avoid the brute force required in Step 3. For example:


Step 5

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Step 3 and Step 4 each show that the only seven possible complete sequences are:

In terms of directly successive part sequences, all those described in Step 2 can contribute to complete sequences apart from 38Sr-31Ga directly preceding 20Ca-2He, and 38Sr-31Ga directly preceding 1H.

Which of Step 3 and Step 4 is better is a matter of personal taste. It is reassuring that they produce the same result.


Copyright July 1999 Henry Bottomley

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