A Mathematician's Holiday - Problem 4: Bags Mix-up.

Requirements:

Three bags with different named labels. A t-shirt or jumper. Of course, you can dress this problem up in many different ways so if you have got a t-shirt or jumper to hand you could have three pencil cases with blue and yellow pencils, for example.

Description:

En route to China our luggage got mixed up. Each of our bags look the same, but the name tags have all been mixed up such that we know that no bag has the correct label (a key point to reiterate). We are allowed to put our hand in a bag of our choice and pull out an object at random (we can look in the bag). Can we identify which bag belongs to each person, from this single piece of information?

You should ensure that the audience understands that we are looking for a definite strategy. We do not need to consider probabilities at all.

Extension:

To be honest, I cannot think of one. This question is pretty much self contained. However, the idea behind this question, i.e. logical deduction, is one of the crucial weapons in the mathematical arsenal. If you have any suggestions for extensions please write them in the comments below.

A Mathematician's Holiday - Problem 3: Late for the Plane.

Requirements:

Some pieces of string, or rope, depending on the size of your demonstration and a play to draw the diagram below in Figure 1. Note that it does not have to be too accurate.

Description:
The critical point to ensure for this question is that you're audience is comfortable with the idea of a straight line between two points being the shortest distance between those two points and, thus, the route of shortest distance.

We are late for our plane and so we want to take the quickest route from the terminal our aeroplane. However, there are two active runways between us and our destination. Since these are dangerous places to be we want to ensure that we minimise our time over the runways and, so, we always run directly across, at right angles to the runway direction. What path minimises our total distance?

Figure 1. The problem set up.

Unless they've seen the problem before it is highly unlikely that your audience will generate the solution shown in the video. However, that does not matter. The fundamental concept we are trying to introduce here is that we are looking for a method, rather than an exact solution. Thus, it should be made clear to the audience that the problem set up does not need to be exact.

Extension
In what case is the shortest route from the terminal to the plane an actual straight line?