I was thinking of how numbers are mentally approximated, and the fact 
that (at least to me) it seems there's a sense in which 1 and 1000 are 
more "mentally distinct" than a billion and a trillion, even though both 
differ by a factor of a thousand and the latter are far more different 
in absolute terms.  A cute and illustrative little math puzzle arose 
from this musing.

Assume exponentiation is right-associative, so a^b^c = a^(b^c).  Now 
consider the two similar-appearing numbers A and B where:

A = 10^10^10^11
B = 10^10^10^10

In a psychological sense these numbers seem pretty similar to me, but 
just how different are they mathematically?  Consider the ratio A/B 
measuring how many times bigger A is than B.  It's interesting to 
consider how accurately you can estimate this ratio without calculating 
it explicitly.  For instance, is it true that A/B is greater than a 
trillion (than is, A is more than a trillion times bigger than B)?

More explicitly, let's write:

A/B = 10^10^10^x

Obviously x is less than 11, and it turns out it's greater than -11. 
It's an interesting experiment to make an intuitive "gut instinct" guess 
as to the correct value of x without calculating anything, and then 
actually calculate it out in full and see how accurate your intuition is.

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