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:

Calculate... \[ 2^{32} \prod_{j=1}^4 \prod_{k=1}^8 \sqrt{ cos^2(j\pi/9) + cos(k\pi/9)^2 } \]

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.