Self-count of Letter-Chunks
ending with « E »
Look at the sequence T1:
T1 = 6, 1, 8, 5, 2, 7, 4, 10, 1, 2, 2, 3, 7, 10, 2, 1, 2, 6, 19, 8, 10, 2,
2, 2, 2, 2, 20, 2, 10, 2, 4, 7, 11, 8, 2, 2, 2, 2, 5, 10, 2, 4, 7, 1, 2, 5, 2, 2,
2, 10, 2, 5, 10, 11, 2, 8, 2, 1, 10, 1, 6, 3, 10, 2, 11, 8, 2, 1, 10, 2, 8, 2,
1, 2, 6, 2, 10, ...
We “translate” T1 in English (capital letters):
T2 = SIX, ONE, EIGHT, FIVE, TWO, SEVEN, FOUR, TEN, ONE,
TWO, TWO, THREE, SEVEN, TEN, TWO, ONE,
TWO, SIX, NINETEEN,
EIGHT, TEN, TWO, TWO, TWO, TWO, TWO, TWENTY, TWO, TEN, TWO, FOUR, SEVEN,
ELEVEN,
EIGHT, TWO, TWO, TWO,
TWO, FIVE, TEN, TWO, FOUR, SEVEN, ONE, TWO, FIVE, TWO, TWO, TWO, TEN, TWO,
FIVE,
TEN, ELEVEN, TWO,
EIGHT, TWO, ONE, TEN, ONE, SIX, THREE, TEN, TWO, ELEVEN, EIGHT, TWO, ONE,
TEN, TWO, EIGHT, TWO,
ONE, TWO, SIX, TWO, TEN, ...
We insert a yellow
stroke immediately after each letter
“E” of T2:
T2 = SIX, ONE|, E|IGHT, FIVE|, TWO,
SE|VE|N, FOUR, TE|N,
ONE|, TWO, TWO, THRE|E|, SE|VE|N, TE|N, TWO, ONE|,
TWO, SIX, NINE|TE|E|N, E|IGHT,
TE|N, TWO, TWO,
TWO, TWO, TWO, TWE|NTY, TWO, TE|N, TWO, FOUR, SE|VE|N, E|LE|VE|N,
E|IGHT,
TWO, TWO, TWO, TWO, FIVE|, TE|N, TWO, FOUR, SE|VE|N, ONE|, TWO, FIVE|, TWO, TWO, TWO, TE|N, TWO, FIVE|,
TE|N, E|LE|VE|N, TWO, E|IGHT,
TWO, ONE|, TE|N,
ONE|, SIX, THRE|E|, TE|N, TWO, E|LE|VE|N, E|IGHT,
TWO, ONE|,
TE|N,
TWO, E|IGHT, TWO,
ONE|, TWO, SIX, TWO, TE|N, ...
Shazam! We notice now that the letter-size of each chunk is given by T1 itself:
T2 = SIX, ONE|, E|IGHT, FIVE|, TWO,
SE|VE|N, FOUR, TE|N,
ONE|, TWO, TWO, THRE|E|, SE|VE|N, TE|N, TWO, ONE|,
6 1 8 5 2 7 4
10 1 2 2
3 7
TWO, SIX, NINE|TE|E|N, E|IGHT,
TE|N, TWO, TWO,
TWO, TWO, TWO, TWE|NTY, TWO, TE|N, TWO, FOUR, SE|VE|N, E|LE|VE|N,
10 2 1 2 6 19 8 10 2 2
2 2 2
E|IGHT,
TWO, TWO, TWO, TWO, FIVE|, TE|N, TWO, FOUR, SE|VE|N, ONE|, TWO, FIVE|, TWO, TWO, TWO, TE|N, TWO, FIVE|,
20
2 10 2 4 7 11
8
TE|N, E|LE|VE|N, TWO, E|IGHT,
TWO, ONE|, TE|N,
ONE|, SIX, THRE|E|, TE|N, TWO, E|LE|VE|N, E|IGHT,
TWO, ONE|,
2 2 2
2 5 10 2 4
7 1 2
5 2 2 2 10
TE|N,
TWO, E|IGHT, TWO,
ONE|, TWO, SIX, TWO, TE|N, ...
2 5 10
11
__________
T2 was built with this simple constraint:
- extend T2 by taking the smallest integer not leading to
a contradiction.
We see thus that T2 cannot start with ONE, or TWO, or THREE, or
FOUR or FIVE (because the presence or the absence of the letter “E” in those
numbers leads to a contradiction). T2 could
start with THIRTY, FORTY, FIFTY and SIXTY also – but SIX is the smallest
available integer. After SIX, we could also have chosen TWENTY:
T3 = SIX, TWE|NTY,
6
... but again, we want the smallest integer fitting the pattern
– which is ONE:
T2 = SIX, ONE|,
6
... ONE fixes the
size of the next chunk, thus:
T2 = SIX, ONE|, E|
6 1
The smallest integer
beginning with “E” in English is EIGHT:
T2 = SIX, ONE|, E|IGHT
6 1
... EIGHT fixes the
size of the next chunk:
T2 = SIX, ONE|, E|IGHT, * * * E|
6 1 8
... The stars stand
for single letters; does TWO fit?
T2 = SIX, ONE|, E|IGHT, T W O, E|
6 1 8
... TWO would fix the
size of the next chunk:
T2 = SIX, ONE|, E|IGHT, TWO, E| * E|
6 1 8 2
... It seems now that
we could extend T2 with ELEVEN:
T2 = SIX, ONE|, E|IGHT, TWO, E| L E| V E N,
6 1 8 2
... ELEVEN fixes the
size of the next chunk;
T2 = SIX, ONE|, E|IGHT, TWO, E| L E| V E N,
6 1 8 2 11
... and we have a problem: the next chunk has size 2 – because
of the third “E” in ELEVEN:
T2 = SIX, ONE|, E|IGHT, TWO, E| L E| V E | N,
6 1 8 2 2
... we have thus a contradiction: the letter-sequence T2 doesn’t match the figures in yellow. We must
work backwards and erase ELEVEN and TWO. Is there a way out? Yes, FIVE seems
ok:
T2 = SIX, ONE|, E|IGHT, * * * E|
6 1 8
T2 = SIX, ONE|, E|IGHT, F I V E|
6 1 8
FIVE dictates the
size of the next chunk, etc.
T2 = SIX, ONE|, E|IGHT, FIVE| * * *
* E|
6 1 8 5
Questions:
- Are the first terms
of T2 correct? They have been computed
by hand...
- Could it be that T2 enters at some point into a loop? Or an
inflating pattern?
- Could someone build
an infinite such sequence?
Best,
É|.
1
__________
[James Dow Allen, on rec.puzzles,
has discovered a loop that
regenerates itself]:
> The sequence (3 2 2 11 3 7 2 2 5 1 2 2 11) if repeated, will regenerate itself, shifted in a
loop of two:
3 2 2 11
3 7 2
2
5 1 2 2 11
..., ELEVEN, THREE, TWO, TWO, ELEVEN,
THREE, SEVEN, TWO, TWO, FIVE, ONE, TWO, TWO, ELEVEN,
THREE, TWO, TWO, ...
..., ELEVE|N THRE|E| TWO, TWO, E|LE|VE|N, THRE|E| SE|VE|N,
TWO, TWO, FIVE| ONE| TWO, TWO, E|LE|VE|N, THRE|E TWO, TWO, ...
5 1
7 2 2 5
1 2 2 11 3 7
2 2 5
...
FIVE ONE
SEVEN TWO TWO
FIVE ONE TWO TWO
ELEVEN THREE SEVEN
TWO TWO
FIVE ...
FIVE| ONE| SE|VE|N TWO TWO FIVE|ONE|TWO
TWO E|LE|VE|N THRE|E| SE|VE|N TWO TWO FIVE|...
3
2
2 11
3 7 2 2 5
1 2 2 11
..., ELEVEN, THREE, TWO, TWO, ELEVEN,
THREE, SEVEN, TWO, TWO, FIVE, ONE, TWO, TWO, ELEVEN,
THREE, TWO, TWO, ...
..., ELEVE|N THRE|E| TWO, TWO, E|LE|VE|N, THRE|E| SE|VE|N,
TWO, TWO, FIVE| ONE| TWO, TWO, E|LE|VE|N, THRE|E TWO, TWO, ...
5 1
7 2 2 5
1 2 2 11 3 7
2 2 5
...
FIVE ONE SEVEN
TWO TWO FIVE ONE TWO TWO ELEVEN
THREE SEVEN TWO TWO FIVE ...
FIVE| ONE| SE|VE|N TWO TWO FIVE|ONE|TWO
TWO E|LE|VE|N THRE|E| SE|VE|N TWO TWO FIVE|...
3
2
2 11
3 7 2 2
5 1 2 2 11
__________
Many thanks, James!
__________
[This
page is the equivalent sequence in French; this one is in French too –
but with the additional constraint that all integers of T2
must be different]