Find the $999^{th}$ digit of
\[ 5^{\left(4^{\left(3^2\right)}\right)} \]
Note: this question is ambiguous. Is it $999^{th}$ digit from the left or right?
You should be able to find both.
Hint: use trunction, or integer division, mod, remainder; some people solved this
by converting the number to a string and extracting the character from the string.
Question #2:
Note that this is the number of tilings of an $8 \times 8$ grid with dominos.
In general there is a formula for the number of tilings of a $m \times n$ grid
with dominos given by
\[ 2^{mn/2} \prod_{j=1}^{m/2} \prod_{k=1}^n \sqrt{ cos^2(j\pi/(m+1)) +
cos^2(k\pi/(n+1)) } \]
To convince yourself that the answer is right you might
want to try the formula with smaller values of $m$ and $n$
that you can calculate by hand. The mathematical symbol $\prod$ represents
product or multiplication.
You do not need to hand this assignment, but I would like you to
introduce yourself and let me know how you did on it. I checked off the
the people that I spoke to, but not everyone on the class list.