Solution to Sheep in Wolf 's Clothing:
Congratulations to Matt Titus and
Jared Nishikawa for solving this one ! The title is alluding to the fact that the
number you are looking at is actually just 1. (So asking for the 2008th digit is
just a red herring.) Here is an edited version of Matt and Jared' s solutions .
Focus first on one of the two cube root terms.
Giving the inside a common
denominator of 216 (nice and cubic), we have
or
In
general, cubing 
will yield another number of
the same form:
Can you find integers a
and a b that make this
expression equal to
It is clear (is it?) that
a would have to be
divisible by 3. Now if you try guessing values for a and b , one of the first
pairs
you'd guess (3 and 1) actually works!
So
Similarly, the
other cube root term is
and their sum is just
1.
Incidentally, this expression is what you get when
you apply the cubic formula to the equation 2x^3-x-1 = 0, demonstrating that
factoring polynomials to find roots (when you can) is way better than using
formulas
