TITLE: Combinatorics in a dynamic system?
QUESTION [5 upvotes]: On page 44 of his article "An Imitation of Life," William Grey Walter claims that "six elements would be more than enough to form a system which would provide a new pattern every tenth of a second for 280 years." What mathematics support this claim? I am only aware of straightforward combinatorics, which states that 6 objects can make 720 unique combinations. How does Walter claim so many more?
https://cse-robotics.engr.tamu.edu/dshell/cs643/papers/walter50imitation.pdf
Thanks!
Kelly

REPLY [3 votes]: I don't understand what Walter is describing concretely in detail, but it sounds vaguely like he is counting the number of possible states a simple "nervous system" composed of $6$ elements can be in, whatever that means (when he talks about "variety of experience" a few sentences later). Here's a count that comes close to his number: among $6$ elements there are ${6 \choose 2} = 15$ pairwise connections, and if we roughly follow his sentence about "six modes" (which I don't understand) let's say that each of these pairwise connections can be in one of $5$ possible states: no connection, a connection $A \to B$, a connection $B \to A$, a connection both ways, and whatever "$A + B$" means. This gives
$$5^{15} \approx 3 \times 10^{10}$$
possible states. His "a tenth of a second for $280$ years" number is $8.8 \times 10^{10}$ so this gets within a factor of $3$. This isn't quite close enough that I can claim that it's what he's talking about, though.