Saturday, 30 April 2016

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?

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