A Mathematician's Holiday - Problem 2: Airport security.

EASY VERSION
Requirements:

Two containers in the ratio 3:5, for example 75ml and 125ml. We used the bottom of drinks bottles and scaled up the quantities in order to make them more visible for a large audience.

Description:

Airports only allow 100ml of fluid through onto a plane. Can we measure out exactly 100ml of fluid using containers that are of the sizes given above and none of the containers have graduation marks? Beyond the trial and error that the audience may try you can actual solve this problem very easily using a graph.

Figure 1 shows a graph illustrating the volume on liquid contained in each. Filling and emptying each container corresponds to horizontal movements, all the way to the left and all the way to the right. Transferring the liquid from one container to the other corresponds with travelling diagonally as far as you can go. By using these two rules you can easily bounce around the graph and successfully reach 100ml.

Figure 1. Solving the fluid problem graphically. The volume of water in the 75ml bottle in along the vertical axis and the volume of water in the 125ml bottle in along the horizontal axis.

Extensions:
In Figure 1 we filled the 125ml bottle first. Can you solve the problem by filling the 75ml bottle first?
What values of fluid can you not possibly make using this method of pouring between the containers?

HARD VERSION
Requirements:

Three containers in the ratio 3:5:8, for example 75ml, 125ml and 200ml.

Description:
The task is the same as above. However, this time we do not have a reservoir of water to fill from, and empty to. We only have a container of 200ml of water, which can be transferred amongst the different sized bottles.

Critically, the solution to this problem depends on ternary coordinates, which are plotted in a triangle form, as seen in Figure 2, see the video for more details. Once the students have seen this form of graph they are able to solve the problem in exactly the same way as the previous question.

Figure 2. Ternary coordinate plot. Each side of the equilateral triangle represents the volume in one of the bottles. Each of the stars represents one of the points on the left. See the video for more details.

Extensions:
Again, what values of fluid can you not possibly make using this method of pouring between the containers?
Instead of starting with 200ml, suppose we started with 125ml only. What fluid volumes can now be made by transferring the fluid between the bottles?

A Mathematician's Holiday - Problem 1: Planning the tour.

Requirements:
Each participant should have a pencil, paper and rubber to draw and modify the diagram shown in Figure 2.

Description: 
In this activity we set the scene of the whole presentation. In particular, we talk about how all the activities were motivated by problems that we faced on our tour. The first problem is then, of course, organising the tour.

As mentioned previously we visited a number of locations around China and South Korea. These locations are shown in Figure 1.

Figure 1. All of the locations we visited on our tour.

Being huge tourists, we didn't want to have to travel between these locations in the same way twice, because each transportation route would allow us to see something different. In Figure 2 we represent each method of transportation by a black line between each of the cities. For example, the O and L mean Oxford and London and the two black lines represent the two different transportations of train and bus.

Figure 2. Representing the city connections.

The challenge is then to find a way around all the cities using all possible forms of transport. We found that we were not able to solve this problem. Can you? Of course the answer is contained in the video at the top.

Extension
From watching the video you should be able to see that identifying whether a set of paths are completely traversable with no repeating is pretty easy. However, suppose we specified the time each path took, how difficult would it be to find the quickest path around all cities?

A Mathematician's Holiday - Prologue.

One of the first activities that got me into outreach was joining Marcus' Marvellous Mathemagicians (M3), which was started in 2008 by Marcus du Sautoy. The idea was that hundreds of schools contact Marcus every year to give presentations to their kids, but unfortunately he simply doesn't have the time to answer all the requests. Thus, myself and a number of other undergraduates and graduates go out to schools and take a number of fun mathematical workshops

Multiple Oxford scientists coming out in force at the Newbury Science Festival

As part of M3, I've been up and down the country giving maths workshops, over to Ireland and Wales, and presented at numerous science festivals across the country.

However, perhaps the best experience I've ever had was when myself, Will Binzi and Dan Martin were invited to tour around the Dulwich Colleges of China and South Korea and demonstrate mathematics to all the kids out there.

From left to right: Dan, Will and myself enjoying the view from the Great Wall.

During our two weeks out there we travelled to Beijing, Suzhou, Shanghai, Zhuhai and Seoul. By the end of the two weeks we were of course exhausted but extremely happy because our workshops had gone down so well.

We presented in classrooms and theatres. The Dulwich college kids were great audiences.

Specifically, because this was a very special tour we decided to write a brand new presentation for the Dulwich students. The presentation was called A Mathematician's Holiday and collected eight different problems that you might come across whilst planning and presenting on such an international tour.

When we came back we decided to film the presentation so it can be used by anybody. The videos can either be found on YouTube, iTunes U or through the University of Oxford's Podcasts page. Over the next 8 weeks I will be presenting each video and discussing any particular techniques that we found aided us in our presenting the workshop.

Crucially, because we were taking this tour on the road, we had to make sure all the props were light, transportable, easily fixable, reproducible, or cheap to buy. In the end we simply used:

• rope;
• plastic bottles;
• bears;
• a t-shirt;
• a jumper;
• three hold alls; and
• a blow up ring.

As a final note on the videos, they're pretty low energy. Normally we would have been giving this presentation in front of 30-50 excitable kids. Here we had 6 sixth-form students and a doctoral researcher, not exactly our prime demographic, but it had to do!

We begin next time by organising the tour!