Determine the total count of positive integers n between 1 and 10^{2007} inclusively such that the last 2007 digits of n and n^{3} are the same. (If n or n^{3} has fewer than 2007 digits, treat it as if it had zeros on
the left to compare the last 2007 digits.)

The answer is definitely at least 2 and probably at least 3.

Aside from 0...01 we also have the 2007 digit automorphic numbers http://oeis.org/A003226 There are usually two for any given digit count, but sometimes only one.