Question 5
Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts for the winter. Jumpy numbers the walnuts from $1$ through $2021$, and digs $2021$ little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.
Prove that there exists a value of $k$ such that, on the $k$th move, Jumpy swaps some walnuts $a$ and $b$ such that $a\lt k\lt b$.
Prove that there exists a value of $k$ such that, on the $k$th move, Jumpy swaps some walnuts $a$ and $b$ such that $a\lt k\lt b$.