The unexpected leading
As I’ve mentioned previously, I’m pretty into swing dancing, having been a teacher with a Brisbane school (Swing Dance Brisbane) for about a year and half. Recently, I’ve been going along to some beginner classes again with friends who are just learning, and I came across an interesting example of paradox come to life.
A really important part of swing dancing is lead and follow - the leads, er, lead a move and the follows should be able to follow it (duh!) without knowing it was coming. The problem is in class, you typically do short routines that makes it easier for both the leads and follows, but you risk losing this spontaneity. So, the teachers told the leads (typically the men) to lead four “basics”. On one of those basics, we were to lead a particular variation we were learning, but the teacher said, “We’re not going to tell you which one to lead it on, because we want it to be a surprise for the lady.”
Uh oh! There’s a problem - if I leave it until the last basic of the four to lead my move, my follow will know it’s coming. Therefore, we both know the move can’t come on the fourth basic. But now, I can’t lead on the third basic either - because my follow knows it can’t come on the fourth, if I don’t lead it on the 1st or 2nd, it would have to come on the 3rd and so they’d know it was coming. Since it has to be a surprise, I can’t lead it on the 3rd basic either. Think about it - it’s true!
By the same logic, I can’t lead it on the 2nd, or even the 1st basic. In short, by this reasoning I can never lead the variation, because it could never be surprise. My follow clearly* came to the same conclusion, and smugly knew she didn’t have to worry about trying to follow a tricky move.
So when I actually lead my variation on the second basic, she was very surprised indeed!
(*) I made that bit up. I was the only mathematician in the room 
This is an example of the unexpected hanging paradox. The simplest form of the paradox would be for the teacher to say “I want you to lead one basic, then lead that tricky move on the second basic and make it a surprise.” The follow would realise that since she knew the move was coming, it couldn’t be a surprise, and hence I wouldn’t lead it. Then, when I lead it, it would be a complete surprise - since she didn’t think it was possible! Thus everything the teacher asked for came true. Woohoo!
I confess I don’t really understand the resolution to the paradox (which originally was about a guy going to be hung unexpectedly , but I think it boils down to your initial assumptions being imprecise in some way - how do you mathematically define “unexpected”?
Either way, yet again there’s mathematics in movement!
A. Reader Says:
I thought you might find this interesting: http://physicsweb.org/articles/world/19/8/2
hehe… Nice.
I posed this one to my students once. Due to the way tutorials were run on any week you had a one-question exam on either calculus or linear algebra. Over the semester they would have been given 5 calculus and 5 linear algebra questions. The exam question for each week was chosen from 4 candidate questions that the students knew (2 calc, 2 LA). The question is: How can I fairly arrange the questions so that they study both lin alg and calculus every week?
It’s a more complicated setup but it works the same. If I blow my 5 calculus questions in the first 5 weeks, they won’t study calculus for the rest of the time. And just like the unexpected hanging, they will know it’s linear algebra or calculus beforehand (by seeing which has 4 and which has 5). So this is a weird hanging paradox that is the plain ol’ one from the last week to the first, but is a “creeping” version from the first week to the fifth.
The students weren’t impressed with this fine logical conundrum. One said: “Like we’d remember things from the first few weeks of work…” Touche, student, touche.
My resolution: choose fortnights randomly so you ruin at most one week (the last) and don’t tell them your scheme