Ice Breaker Card Tricks

Today was the first day of classes for the spring semester. I have a light teaching load this term, which is my reward for having an especially heavy teaching load last term. Just two classes, and they both meet in the afternoon, no less. For a night-owl like me that's a good deal.

One of my courses is second-semester calculus. Since this is the continuation of a course I taught last term, I get to see a lot of familiar faces. That's always nice, since it usually takes me a while to learn everyone's name. On tap are lots of exponential functions, logarithms, and trigonometry. Also, I'm sure we'll integrate a few things before the end of the term.

I'm also teaching an upper-level course called: The Real Number System. This is a course intended for math majors, in which we focus on how to construct the various numbers systems. Basically, we try to answer questions that are typically just ignored in other classes. For example, what could it possibly mean to raise a number to an irrational power? For that matter, how do you add two irrational numbers? They're infinite, non-repeating decimals after all. When you enlarge the counting numbers to the set of integers, why must it be the case that the product of two negatives is a positive? Along the way we discuss things like the Zermelo-Frankel axioms for set theory, the Peano axioms for the natural numbers, and various other esoteric topics. It's a course that can often seem pretty abstract, but is actually good preparation for both real analysis and abstract algebra, both of which are required courses for math majors.

I like to get things started slowly, so on the first day of class, in addition to going over the syllabus, of course, I like to do a couple of mathematical card tricks. The first one is fairly difficult, and requires some practice to perform fluidly, but the effect is astonishing.

You start with the ten, jack, queen, king and ace of every suit, twenty card total, thoroughly shuffled by the audience. You ask an audience member for their favorite card suit. Let's say they reply hearts. The magician then launches into a discourse on chaos theory, discussing how spontaneous order can sometimes arise in seemingly chaotic systems. The card are dealt into a pile two at a time, with the audience allowed to make certain decisions about which cards are face up and which are face down. You then go through the deck again, this time by fours, with audience members allowed to decide whether the cards are left alone or flipped over en masse. The cards are then dealt onto a table in five rows of four, with the audience allowed to decide whether the cards are dealt left to right or right to left in each row. If a smart aleck tells you to deal from the center out, that can be accommodated. You then “fold up&rduo; the deck by having the audience successively select from among the four edges of the rectangular array of cards, and then flipping the cards over that edge. After the deck is completely reassembled, the cards are spread on the table. It is seen that only five cards remain face down. They are flipped over and revealed to be the five hearts.

I realize that's hard to follow, so here's a YouTube video of someone performing three versions of a very similar trick:

You can always pick out the students who are going to give you trouble during the term. They are the ones who are completely unimpressed with this trick.

If you just have to know how it's done, you can have a look at this explanation, by Arthur Benjamin of Harvey Mudd College.

Since I already have the ten, jack, queen, king and ace of every suit at hand, I now launch into a second, much simpler, trick. The cards are divided into four piles by suit. The magician then explains the concept of “immiscible liquids” and says he is going to illustrate the principle using playing cards. Oil and water can be forced to mix, by putting them in a bottle and shaking them up, but once left at rest they quickly separate. Likewise here. The tens, jacks, queens, kings and aces are currently being forced to mix, but their natural tendency is to separate. The cards are then assembled into one pile. They are cut and shuffled. An audience member is then allowed to cut the cards. The cards are then dealt face down into five piles of four cards each. When the piles are turned over we find that all the tens are together, all the jacks are together and so on.

Here's a basic version of the trick:

The performance can be made more spectacular by adding various false shuffles, or false triple cuts.

My usual joke is that at some point during the term I will tell them how I did those tricks, but they will not know ahead of time which day that will be. That means they will have to come to class every day, just in case that's the day I reveal how the trick is done. Mostly, though, I do these tricks as an ice breaker. Who talks about hard math with people they just met?