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.
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.
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)
92U-73Tawhich 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-64Gdwhich could come
from 72Hf-68Er or 30Zn-21Sc, and lead to 63Eu-62Sm or 20Ca-2He
63Eu-62Smwhich could come
from 67Ho-64Gd or 38Sr-31Ga, and lead to 61Pm-39Y, 30Zn-21Sc or
20Ca-2He
61Pm-39Ywhich could come from
72Hf-68Er or 63Eu-62Sm, and lead to 92U-73Ta or 38Sr-31Ga
38Sr-31Gawhich could come
from 61Pm-39Y and lead to 63Eu-62Sm, 30Zn-21Sc, 20Ca-2He or 1H
30Zn-21Scwhich could come
from 63Eu-62Sm or 38Sr-31Ga and lead to 67Ho-64Gd, 20Ca-2He or 1H
20Ca-2Hewhich 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
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.
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:
A sequence starting with 92U-73Ta directly preceding 1H,
or starting with 38Sr-31Ga directly preceding 1H, would
not be complete.
If a complete sequence starts with 92U-73Ta then later 61Pm-39Y
must directly precede its only other outlet 38Sr-31Ga.
If a complete sequence starts with 92U-73Ta 20Ca-2He,
it must continue directly with 72Hf-68Er, which has no
other available source. Later 67Ho-64Gd must directly
precede its only other outlet 63Eu-62Sm. 30Zn-21Sc then
cannot directly precede 67Ho-64Gd 63Eu-62Sm, since
if it did both 92U-73Ta 20Ca-2He 72Hf-68Er and
30Zn-21Sc 67Ho-64Gd 63Eu-62Sm would impossibly
each directly precede their only remaining outlet 61Pm-39Y 38Sr-31Ga.
So 92U-73Ta 20Ca-2He 72Hf-68Er must directly
precede 67Ho-64Gd 63Eu-62Sm (its only remaining
source) and 30Zn-21Sc must directly precede its only
remaining outlet 1H. This only leaves one remaining
source and outlet for 61Pm-39Y 38Sr-31Ga. So if a
complete sequence starts with 92U-73Ta 20Ca-2He, it
must be 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm
61Pm-39Y 38Sr-31Ga 30Zn-21Sc 1H.
If a complete sequence starts with 92U-73Ta 72Hf-68Er,
then 20Ca-2He must directly precede 1H, and must
therefore be at the end of the sequence.
A complete sequence which starts with 92U-73Ta 72Hf-68Er
cannot finish 38Sr-31Ga 20Ca-2He 1H. If it did,
then the end would have to be 61Pm-39Y 38Sr-31Ga 20Ca-2He 1H.
92U-73Ta 72Hf-68Er would have to directly precede 67Ho-64Gd
to enable completeness, and this in turn would have to
directly precede its only remaining outlet 63Eu-62Sm.
There would then be no outlet for 30Zn-21Sc.
A complete sequence which starts with 92U-73Ta 72Hf-68Er
must therefore end 67Ho-64Gd 20Ca-2He 1H, 63Eu-62Sm 20Ca-2He 1H
or 30Zn-21Sc 20Ca-2He 1H.
If a complete sequence starts with 92U-73Ta 72Hf-68Er
and ends with 67Ho-64Gd 20Ca-2He 1H, then 92U-73Ta 72Hf-68Er
must directly precede 61Pm-39Y 38Sr-31Ga to ensure
completeness. 30Zn-21Sc is then the only remaining source
for 67Ho-64Gd 20Ca-2He 1H and 38Sr-31Ga is the
only remaining source for 63Eu-62Sm. So if a complete
sequence starts with 92U-73Ta 72Hf-68Er and ends
with 67Ho-64Gd 20Ca-2He 1H, it must be 92U-73Ta
72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd
20Ca-2He 1H
If a complete sequence starts with 92U-73Ta 72Hf-68Er
and ends with 63Eu-62Sm 20Ca-2He 1H, then 92U-73Ta 72Hf-68Er
must directly precede 61Pm-39Y 38Sr-31Ga as its only
remaining source. 38Sr-31Ga is the only remaining source
for 30Zn-21Sc, which in turn must directly precede its
only remaining outlet 67Ho-64Gd. So if a complete
sequence starts with 92U-73Ta 72Hf-68Er and ends
with 63Eu-62Sm 20Ca-2He 1H, it must be 92U-73Ta
72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm
20Ca-2He 1H
If a complete sequence starts with 92U-73Ta 72Hf-68Er
and ends with 30Zn-21Sc 20Ca-2He 1H, then 67Ho-64Gd
must directly precede its only remaining outlet 63Eu-62Sm.
61Pm-39Y 38Sr-31Ga must directly precede its only
remaining outlet 30Zn-21Sc 20Ca-2He 1H. So if a
complete sequence starts with 92U-73Ta 72Hf-68Er and
ends with 30Zn-21Sc 20Ca-2He 1H, it must be 92U-73Ta
72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc
20Ca-2He 1H
A complete sequence must start with 92U-73Ta 20Ca-2He
(in which case 38Sr-31Ga directly precedes 30Zn-21Sc),
with 92U-73Ta 72Hf-68Er (in which case 20Ca-2He
directly precedes 1H), or with 38Sr-31Ga (in which case
38Sr-31Ga does not directly precede 1H); therefore there
are no complete sequences where 38Sr-31Ga directly
precedes 1H.
If a complete sequence starts with 38Sr-31Ga then later
61Pm-39Y must directly precede its only other outlet 92U-73Ta.
A complete sequence cannot start with 38Sr-31Ga 20Ca-2He.
If it did then either this would then directly precede 1H
(and so would not be complete) or directly precede its
only other outlet 72Hf-68Er. 61Pm-39Y 92U-73Ta would
then only have one remaining outlet, 1H. 63Eu-62Sm would
be the only remaining source for 30Zn-21Sc, with 67Ho-64Gd
its only remaining outlet; this in turn would have to
directly precede its only remaining outlet 63Eu-62Sm,
leading to an impossible loop.
If a complete sequence starts with 38Sr-31Ga 63Eu-62Sm
then this is the only remaining source for 30Zn-21Sc. 72Hf-68Er
would then be the only other source for 61Pm-39Y 92U-73Ta,
and 20Ca-2He would then be the only source for 72Hf-68Er
which did not create a loop. 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc
would be the only remaining source for 67Ho-64Gd. 92U-73Ta
would be the only remaining source for 1H. So a complete
sequence starting with 38Sr-31Ga 63Eu-62Sm must be 38Sr-31Ga
63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y
92U-73Ta 1H.
If a complete sequence starts with 38Sr-31Ga 30Zn-21Sc
then this must directly precede 67Ho-64Gd or 20Ca-2He. 67Ho-64Gd
is then the only remaining source for 63Eu-62Sm.
If a complete sequence starts with 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd
then it will directly precede 63Eu-62Sm. 72Hf-68Er must
directly precede its only other outlet 61Pm-39Y 92U-73Ta,
and to avoid creating a loop, must be preceded by 20Ca-2He.
63Eu-62Sm must directly precede its only remaining outlet
20Ca-2He. 92U-73Ta is then the only remaining source for
1H. So if a complete sequence starts with 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd
then the complete sequence must be 38Sr-31Ga 30Zn-21Sc
67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta
1H
If a complete sequence starts with 38Sr-31Ga 30Zn-21Sc 20Ca-2He
then this must directly precede 72Hf-68Er to enable
completeness. 67Ho-64Gd 63Eu-62Sm must directly
precede its one remaining outlet 61Pm-39Y 92U-73Ta,
so 72Hf-68Er has one remaining outlet, 67Ho-64Gd, and 92U-73Ta
can only precede 1H. So if a complete sequence starts
with 38Sr-31Ga 30Zn-21Sc 20Ca-2He then the
complete sequence must be 38Sr-31Ga 30Zn-21Sc 20Ca-2He
72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 1H
The seven identified complete sequences are indeed
complete, and they are therefore the only complete
sequences.
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.