Question 59
A thimblerigger has 2021 thimbles numbered from 1 through 2021. The thimbles are arranged in a circle in arbitrary order. The thimblerigger performs a sequence of 2021 moves; in the $k^{\text{th}}$ move, he swaps the positions of the two thimbles adjacent to thimble $k$.
Prove that there exists a value of $k$ such that, in the $k^{\text{th}}$ move, the thimblerigger swaps some thimbles $a$ and $b$ such that $a \lt k \lt b$.
Prove that there exists a value of $k$ such that, in the $k^{\text{th}}$ move, the thimblerigger swaps some thimbles $a$ and $b$ such that $a \lt k \lt b$.