~~~ ~~~ From:

~~~ ~~~ Subject:

Allan C. Wechsler writes:

> It couldn't be very pointy. From the most distant configuration,

> there are 6 positions immediately before it. There are 6^2 two steps

> away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually.

>

>Very good. This is a necessary insight, regardless of the exact

>numerical details. (For example, you mean 12, not 6.) But the

>possible flaw is that there might be more than one maximally distant

>state; if their sets of neighbors overlap viciously enough, this

>effect could make the tail pointier. You can make valence-12 graphs

[deletia]

All this misses the point (so to speak) which is that 12^N is _exceedingly_

pointy for our purposes. If one samples only 1000 positions out of ~10E19,

then one could very well miss a 12^N tail of length 14 moves!

The estimate of 22 as an upper limit relies on the intuition that

the distribution is MUCH blunter than this.

Dave Ring

dwr2560@zeus.tamu.edu