[seqfan] RE : Re: Glass worms

Eric Angelini Eric.Angelini at kntv.be
Tue Mar 10 00:29:18 CET 2009

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 <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,





De: seqfan-bounces at list.seqfan.eu de la part de Mitch Harris
Date: lun. 09/03/2009 19:48
À: 'Sequence Fanatics Discussion list'
Objet : [seqfan] Re: Glass worms

> Take a finite row of glasses, each one containing an
> interger number of liquid-units (zero = empty glass)
> Procedure:
> - take the leftmost glass,
> - consider the number k of liquid-units it contains,
> - empty the glass in the k-th glass on it's right,
> - put the now empty glass at the right end of the row;
> - start the procedure again.

Your procedure is vey reminiscent of (but not identical to) siteswap
notation for juggling patterns:

http://www.research.att.com/~njas/sequences/?q=siteswap&sort=0&fmt=0&languag <http://www.research.att.com/~njas/sequences/?q=siteswap&sort=0&fmt=0&languag> 

explanation at

  http://www.juggling.org/bin/mfs/JIS/help/siteswap/ <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).



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