This was on my grade 10 Cayley math contest a few days ago. It is really interesting, but super hard.

We have a circle with 10 points (think of a clock that goes from 0-9). The dial starts at 0. In each move Tom moves the dial $\displaystyle n^n$ times. So on the first move Tom moves $\displaystyle 1^1$ times, and the dial is at 1. Second move Tom moves $\displaystyle 2^2$ times, and the dial is at 5, and third move Tom moves $\displaystyle 3^3$ times, and the dial is at 2. What number would the dial be pointing at after 1234 moves?

Essentially it is asking what $\displaystyle \sum_{n=1}^{1234} n^n \text { mod } 10 $ is.