Of Baby Hummers and clock arithmetic with Aryabhata and Archimedes

I spent a pleasant evening at the National Museum of Mathematics  this week – the first session of a semester long program of lecture demonstrations about mathematics and magic. The instructor is Manjul Bhargava, the famous Princeton mathematician. I thought the ideas were worth discussing in a more public forum, so I resolved to demonstrate my own version of the cool tricks he showed. All the tricks have some cute mathematical nugget hidden inside and he spent the last ten minutes basically revealing the secret behind the trick.

I have changed the tricks slightly to prevent a complete copy of what he presented, but the essence of the idea is exactly the same.

The first is called a Baby Hummer card trick, a creation of Bob Hummer. I demonstrate it as well as a ten-card Hummer trick in the following videos.

Here’s the third trick (this is a variation of what I saw at the class) : I invoke my favorite mathematicians, Aryabhata and Archimedes and the magic they wove.

Why do these three tricks work?

Let’s look at the first two.

The Hummer “arrangement” and its preservation under the Hummer “maneuver” is key. It is described in the following pictures

Now add one “error” card in the arrangement

In my tricks, I basically put your selected card in as an “error” in the Hummer arrangement.

The Hummer “maneuver” also protects two or more “error” cards, which explains some of the variants of the puzzle you could find on YouTube.

The third trick is easy to think about and uses the arithmetic of a clock – also called modulo arithmetic.  I will leave it at that.

Thanks to my videographer and collaborator Rajeswari Satish. And thanks to the National Museum of Mathematics for organizing interesting events nearly every week.