Date: Mon, 9 Mar 2009 17:07:10 +0100
From: "Eric Angelini"
To: "Sequence Fanatics Discussion list"
Subject: [seqfan] Glass worms
Take a finite row of glasses, each one containing an
integer number of liquid-units (zero being allowed)
Procedure:
- take the leftmost glass,
- consider the number k of liquid-units it contains,
- empty the glass in the k-th glass on its right,
- put the now empty glass at the right end of the row;
- start the procedure again.
Example :
\ 1 / \ 2 / \ 4 / \ 1 / \ 0 /
\_/ \_/ \_/ \_/ \_/
\ 0 / \ 3 / \ 4 / \ 1 / \ 0 /
\_/ \_/ \_/ \_/ \_/
\ 3 / \ 4 / \ 1 / \ 0 / \ 0 /
c) \_/ \_/ \_/ \_/ \_/
\ 0 / \ 4 / \ 1 / \ 3 / \ 0 /
d) \_/ \_/ \_/ \_/ \_/
\ 4 / \ 1 / \ 3 / \ 0 / \ 0 /
e) \_/ \_/ \_/ \_/ \_/
\ 0 / \ 1 / \ 3 / \ 0 / \ 4 /
f) \_/ \_/ \_/ \_/ \_/
\ 1 / \ 3 / \ 0 / \ 4 / \ 0 /
g) \_/ \_/ \_/ \_/ \_/
\ 0 / \ 4 / \ 0 / \ 4 / \ 0 /
h) \_/ \_/ \_/ \_/ \_/
\ 4 / \ 0 / \ 4 / \ 0 / \ 0 /
i) \_/ \_/ \_/ \_/ \_/
\ 0 / \ 0 / \ 4 / \ 0 / \ 4 /
j) \_/ \_/ \_/ \_/ \_/
\ 0 / \ 4 / \ 0 / \ 4 / \ 0 /
k) \_/ \_/ \_/ \_/ \_/
this configuration being the same as (h)
Questions :
Should we represent a glass-configuration as a string of
characters, the above sequence would look like:
(a) 12410 --> (b) 03410 --> (c) 34100 --> (d) 04130 -->
(e) 41300 --> (f) 01304 --> (g) 13040 --> (h) 04040 -->
(i) 40400 --> (j) 00404 --> (k) 04040 --> (h) (loop)
What would be the seq. W(1) of integers (like 12410 or 13040)
which end in a loop?
What would be the seq. W(2) of the smallest integers part
of a loop (like 40400) [We can see those integers as worms
(glasses-worms) moving to the right in successive moltings
-- thus the name of the (french) page here, with the word-
play vers/verre (worm/glas):
http://www.cetteadressecomportecinquantesignes.com/GlassWorms.htm
Best,
É.
From: "Mitch Harris"
To: "'Sequence Fanatics Discussion list'"
Date: Mon, 9 Mar 2009 14:48:23 -0400
Your procedure is vey reminiscent of (but not identical to) siteswap
notation for juggling patterns:
http://www.research.att.com/~njas/sequences/?qe
explanation at
http://www.juggling.org/bin/mfs/JIS/help/siteswap/
But rather than thinking of you glassworm sequences about numbers in decimal
(a much too contingent selection), think about number of unique patterns
(with restrictions like max amount in a glass).
...
> Should we represent a glass-configuration as a string of
> characters, the above sequence would look like:
> (a) 12410 --> (b) 03410 --> (c) 34100 --> (d) 04130 -->
> (e) 41300 --> (f) 01304 --> (g) 13040 --> (h) 04040 -->
> (i) 40400 --> (j) 00404 --> (k) 04040 --> (h) (loop)
>
> What would be the seq. W(1) of integers (like 12410 or 13040)
> which end in a loop?
Isn't it obvious that they all end in a cycle? Or do you expect to encode a
TM-like thing?
(they all have to loop eventually, because you have a finite amount of
liquid so the # of glasses is finite).
Mitch
From: "Jacques Tramu"
To: "Sequence Fanatics Discussion list"
Date: Mon, 9 Mar 2009 23:36:14 +0100
We could also consider the worst worms case :
What is the starting configuration which leads to the larger number of
moves,
before detecting a cycle ?
I found the following (not exhaustive search)
sequence for glasses
number of glasses , {start config} , max moves to cycle detection
1 {0} 1
2 { 0, 1} 2
3 { 1, 0, 1} 4
4 { 0, 1, 1, 1} 7
5 { 0, 1, 0, 2, 1} 10
6 { 3, 0, 0, 0, 0, 2} 13
7 { 2, 0, 0, 5, 4, 4, 1} 20
8 { 0, 0, 0, 2, 2, 2, 0, 1} 22
9 { 1, 5, 1, 1, 1, 2, 0, 0, 3} 42 (!)
10 { 3, 0, 4, 2, 1, 0, 0, 1, 0, 3} 43
11 { 0, 3, 0, 1, 4, 0, 5, 10, 1, 1, 3} 63
Example :
{ 1, 0, 1}
{ 1, 1,0} move 1
{ 2,0,0} move 2
{ 0,2,0} move 3
{ 2,0,0} move 4 <-- cycle detection
Regards,
JT
From: "Eric Angelini"
To: "Sequence Fanatics Discussion list"
Thank you for yr remarks, Mitch -- and the interesting links provided.
My idea came after the re-discovery of this sequence, two days ago:
http://www.research.att.com/~njas/sequences/A137417
>they all end in a cycle
... no, 3 or 14 or 125 die (if an integer m has at least one digit >= the length of m, death is the result -- as you cannot empty a glass elsewhere than in an existing glass -- I should have mentionned that rule)
... you are right for the other integers -- they all loop [but still W(1) and W(2) would be interesting to compute, I guess]
... I like Jacques Tramu's remark about the "worst cycles" too!
... and I think we should invent a rule for glasses having more than 9 liquid-units (those glasses cannot be part of an integer as the string of characters would be ambiguous -- the number of characters being > to the number of glasses)
Thanks again for yr msg,
Best,
E.
Date: Tue, 8 Sep 2009 00:33:55 +0000 (UTC)
From: "Joseph S. Myers"
The message
http://list.seqfan.eu/pipermail/seqfan/2009-March/001066.html
giving the figures for A151986 says "not exhaustive search", which I
interpret as indicating that the values may be conjectural (maximum found,
rather than known to be maximum possible). My program (finding all
possible cycles then searching backwards from those for long paths leading
to a cycle) agrees with the given figures up to 10 glasses, but I get 64
not 63 for 11. The following are the figures and (not necessarily unique)
worst examples according to my program; can someone at least confirm these
examples do give the indicated figures as lower bounds (so I'm
interpreting the problem in the intended way)?
1 glasses: max 1: 0
2 glasses: max 2: 0 1
3 glasses: max 4: 0 1 1
4 glasses: max 7: 0 0 2 1
5 glasses: max 10: 0 0 0 3 1
6 glasses: max 13: 0 0 0 3 0 2
7 glasses: max 20: 0 0 0 6 0 3 1
8 glasses: max 22: 0 0 0 0 0 4 2 1
9 glasses: max 42: 0 0 0 0 0 5 0 6 3
10 glasses: max 43: 0 0 0 0 0 0 5 0 6 3
11 glasses: max 64: 0 0 0 0 0 8 0 5 3 0 2
12 glasses: max 65: 0 0 0 0 0 0 8 0 5 3 0 2
13 glasses: max 146: 0 0 0 0 0 12 0 8 0 5 3 0 2
14 glasses: max 147: 0 0 0 0 0 0 12 0 8 0 5 3 0 2
15 glasses: max 203: 0 0 0 0 0 0 0 0 0 8 6 12 3 0 5
16 glasses: max 239: 0 0 0 0 0 0 0 12 0 9 6 0 5 0 3 1
17 glasses: max 412: 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1
18 glasses: max 413: 0 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1
19 glasses: max 414: 0 0 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1
20 glasses: max 415: 0 0 0 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1