Hypercups: Flipping Cups With More Than Two Sides

In their 2010 article entitled ``How to invert $n$ cups $m$ at a time" in Mathematics Today, Man-Keung Siu and Ian Stewart extend a classic trick in which $3$ cups are flipped $2$ at a time to any number of cups, flipped any number at a time.

Here, we generalize another feature of the problem.

A typical cup in our universe, as far as we know, has two ``sides", right side up and upside down.

But what if the cup had $k$ ``sides" with $k\ge 2$?

We call these $k-$hypercups.

The classic trick depends on parity.

In this article, we show the analogous hypercups trick depends on greatest common divisors.

We generalize all of Siu and Stewart's results to $k-$hypercups.

Our results imply the known results for $2-$hypercups, also known as cups.