## 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

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

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
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

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

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.

• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 63Eu-62Sm (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He 1H Success
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 1H (incomplete)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 67Ho-64Gd 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 92U-73Ta (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 61Pm-39Y (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 1H Success
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 1H Success
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 20Ca-2He 72Hf-68Er (duplication)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 20Ca-2He 1H (incomplete)
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 1H (incomplete)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 63Eu-62Sm (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 1H Success
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 1H (incomplete)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 61Pm-39Y (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 1H (incomplete)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 20Ca-2He (duplication)
• 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga 1H (incomplete)
• 92U-73Ta 20Ca-2He 1H (incomplete)
• 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm (duplication)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er 67Ho-64Gd 20Ca-2He 72Hf-68Er (duplication)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er 67Ho-64Gd 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er 61Pm-39Y (duplication)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm (duplication)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 20Ca-2He (duplication)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y (duplication)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 67Ho-64Gd (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H Success
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 20Ca-2He (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm (duplication)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er 67Ho-64Gd 20Ca-2He (duplication)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 63Eu-62Sm 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er 67Ho-64Gd (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er 61Pm-39Y (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 72Hf-68Er 61Pm-39Y (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 67Ho-64Gd (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H Success
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 67Ho-64Gd (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 1H Success
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 20Ca-2He (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 20Ca-2He (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 30Zn-21Sc 1H (incomplete)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 20Ca-2He (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 30Zn-21Sc 1H (incomplete)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 20Ca-2He (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 67Ho-64Gd 20Ca-2He (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 72Hf-68Er (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 20Ca-2He (duplication)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H (incomplete)
• 38Sr-31Ga 20Ca-2He 72Hf-68Er 61Pm-39Y 38Sr-31Ga (duplication)
• 38Sr-31Ga 20Ca-2He 1H (incomplete)
• 38Sr-31Ga 1H (incomplete)

### Step 4

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.

### Step 5

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

• 92U-73Ta 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 20Ca-2He 1H
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 1H
• 92U-73Ta 72Hf-68Er 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 1H
• 92U-73Ta 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 38Sr-31Ga 30Zn-21Sc 1H
• 38Sr-31Ga 63Eu-62Sm 30Zn-21Sc 67Ho-64Gd 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H
• 38Sr-31Ga 30Zn-21Sc 67Ho-64Gd 63Eu-62Sm 20Ca-2He 72Hf-68Er 61Pm-39Y 92U-73Ta 1H
• 38Sr-31Ga 30Zn-21Sc 20Ca-2He 72Hf-68Er 67Ho-64Gd 63Eu-62Sm 61Pm-39Y 92U-73Ta 1H

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.