Broady The Boar is playing a boring board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, ... $2020$ in that order clockwise. Broady is rolling $2020$-sided die which randomly produces a whole number between $1$ and $2020$, inclusive.
    
    Broady starts at the point labelled $0$. After each dice roll, Broady moves up the same number of points as the number rolled (point $2020$ is followed by point $0$). For example, if they are at $0$ and roll a $5$, they end up at $5$. If they are at $2019$ and roll a $3$, they end up at $1$.
    
    Broady continues rolling until they return to the point labelled $0$. What is the expected number of times they roll the dice?


[i]2021 CCA Math Bonanza Lightning Round #2.3[/i]
1. Let \( E \) be the expected number of moves it takes for Broady to return to point \( 0 \) after starting at a nonzero point. 

2. If Broady is at a nonzero point, there is a \(\frac{1}{2020}\) chance that the next roll will bring Broady back to point \( 0 \). This is because there are 2020 possible outcomes, and only one of them will result in landing on point \( 0 \).

3. Therefore, the expected number of moves \( E \) can be expressed as:
   \[
   E = 1 + \frac{1}{2020} \cdot 0 + \frac{2019}{2020} \cdot E
   \]
   Here, \( 1 \) represents the move just made, \(\frac{1}{2020} \cdot 0\) represents the case where Broady returns to \( 0 \) immediately (which contributes \( 0 \) additional moves), and \(\frac{2019}{2020} \cdot E\) represents the case where Broady does not return to \( 0 \) and thus expects to make \( E \) more moves.

4. Simplifying the equation:
   \[
   E = 1 + \frac{2019}{2020} \cdot E
   \]
   \[
   E - \frac{2019}{2020} \cdot E = 1
   \]
   \[
   \left(1 - \frac{2019}{2020}\right) E = 1
   \]
   \[
   \frac{1}{2020} E = 1
   \]
   \[
   E = 2020
   \]

5. Since Broady starts at point \( 0 \) and leaves on the first roll, the total expected number of rolls is:
   \[
   1 + E = 1 + 2020 = 2021
   \]

The final answer is \(\boxed{2021}\)