Abstract: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.
From: Lauren Keough [view email]
[v1]
Thu, 25 Jun 2026 15:18:48 UTC (802 KB)
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