==> pickover/pickover.04.p <==
Title: Cliff Puzzle 4: Time in a Bottle
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name,
address, affiliation, e-mail address.  If you like, tell me a little bit
about yourself.  PLEASE ALSO directly mail me a copy of your response
in addition to any responding you do in the newsgroup.  I will assume it
is OK to describe your answer in any article or publication I may write
in the future, with attribution to you, unless you state otherwise.
Thanks, Cliff Pickover

* * *

Consider a chain of bottles (B) each connected to one another by a thin
tube. A marble is placed in bottle 1.
Each tube contains a one-way valve so marbles can only
go from left to right in the tubes which are symbolized with "-" marks:

1   2   3   4
B - B - B - B -


The tubes are thin so it takes
1 hour of constant random shaking to get the marble from B1 to B2.
Likewise for each bottle.

I have not fully described the bottle collection.  Each bottle
has a backward 1-way tube to bottle 1.  I've tried to diagram these
with "*" symbols.  Each time the marble enters bottle B(N) it has
a 50% probability of going back to bottle 1 via these tubes.


****<********
*           *
***<*****   *
*       *   *
* * *   *   *
1   2   3   4
B - B - B - B -

Stop And Think

1.  In how many hours will you expect to get the marble out of bottle 10
after placing the marble in bottle 1?

2. Is there a general formula for the amount of time
required to get the ball out of bottle N into bottle N+1 given
a probability P of backwards motion (given as 50% in this problem)?

3.  In how many hours will you expect to get the marble out of bottle 10
after placing the marble in bottle 1 given two backward tubes for each
bottle instead of one backward tube?