During high school I participated in a few math contests, and in undergrad I wrote several problems for the Caltech Harvey Mudd Math Competition and USA Math Talent Search. Every two weeks I will post a recreational (non-research) math problem which I encountered during this time (and particularly enjoyed) below.

A thousand coins, labeled $1$ through $1000$, are arranged in a line in some order. We say two coins labeled with integers $i$ and $j$ form an inversion if $i < j$, but coin $i$ appears to the right of coin $j$ in the linear arrangement.

Consider the following procedure: for each $i = 1, 2, \dots, 1000$ in that order, we move coin $i$ from its current position to its opposite, position reflected in the middle of the sequence (i.e., if coin $i$ currently has exactly $c$ coins to its left, then we place it in a new position so that it now has exactly $c$ coins to its right). Prove that after this procedure is completed, the new arrangement has the same number of pairs of coins forming inversions that the original ordering had.

A new problem might be posted here on March 20th, 2023.

Previously posted problems can be found here.