Questions Regarding Statistical Chamber Assemblies in Clifford Pickover's Book TIME: A TRAVELER'S G

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Questions Regarding Statistical Chamber Assemblies in Clifford Pickover\'s Book TIME: A TRAVELER\'S GUIDE

Hi everyone:

In Chapter 16 of Clifford Pickover's book TIME: A TRAVELER'S GUIDE, he discusses scenarios involving sets of concatenated "chambers" that have both one-way-forward and one-way-backward connectors (except, of course, for the first and last chambers, which, respectively, only have one-way-forward and one-way-backward connectors). In these scenarios, he considers how long it would take a marble to get from the first chamber to the last chamber if the set was shaken up to get the marble out. Assuming that it would take one hour to shake out a marble from one of these such chambers with only one opening, he suggests that it would take approximately...



...hours to shake the marble out of a set of n chambers each (except for the first chamber, of course) having M one-way-backward valves. He also seems to use the notation C(n,M) as shorthand to describe such a system.

All this being said, I'm not sure how he arrived at some of the numbers that he arrived at. Here is a summary of the times he arrived at for various chamber assemblies...

Chamber Assembly Time
---------------- --------------------------
C(9,2) a little over a year
C(9,5) human life span
C(15,5) about 1.6 million years
C(15,7) about 230 million years
C(15,10) over 9 billion years

Although the formula seemed to work reasonably well for C(9,2) and C(9,5) and even C(15,10), it didn't seem to work well at all for C(15,5) or C(15,7).

So, was I wrong, was Pickover wrong, or were both of us wrong? Also, in one place he suggests that the connectors that aren't one-way-backward are "free-flowing", whereas in another place he seems to suggest that they are "forward"; which one, if either, is consistent with the scenarios presented in this chapter?

Thanks in advance for any help!