Saturday, 25 June 2016

A Mathematician's Holiday - Problem 6: The Tiny Lift.

We use two beards to go along with the story that we have concocted. However, you can use two hats, two name badges, etc. Anything that will denote two objects as being similar.
Once we got into the hotel we found that the lift to our floor was very small. This meant that only  one of us could go up to the first floor at once. This leads to a problem: you see Will and I have bonded over our beards, whilst Dan does not have any facial hair! Thus, neither Will nor I want to be left alone with Dan because we would have nothing to discuss. How do you get us all from the ground floor to the first floor, without ever leaving an hirsute person with a beardless person?

Figure 1. Problem set up.
This problem is simply the fox, chicken, grain problem, a classic brain teaser that has been around for centuries. The audience should have no problem producing an adhoc solution in no time. However, there are two key factors that should be extracted. Firstly, having them demonstrate the problem is a very fun, visual and hilarious thing to do, if you are using beards and, so, it rejuvenates a flagging audience. The second (mathematical) point of the solution to extract is that the audience probably solved the problem by trial and error, however, the solution can be solved neatly by using a graph, just like the "airport security" problem.

Extension 1
Can you solve these two related, but different transport problems:
Vampires and maidens
Three maidens and three vampires must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are maidens present on the bank, they cannot be outnumbered by vampires (otherwise the vampires would bite the maiden).
Jealous husbands
Three married couples want to cross the river in a boat that can only hold two people. Unfortunately, no woman can be in the presence of another man unless her husband is also present.

Extension 2
How is the vampire and maidens puzzle related to the Jealous husbands puzzle?

Saturday, 11 June 2016

A Mathematician's Holiday - Problem 5: Hotel Fire!

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.
Having got to the hotel we unfortunately find that it is on fire. Thankfully, there is a river very close to our location, where we can load up buckets with water and, thus, help put the fire out. What is the quickest route from our location, to the river and to the hotel?
Figure 1. Problem set up.

This problem is very similar to "Late for the Plane" and works well if they are done in combination with one another. Again, the question relies on using the ropes to measure distance, in order to measure time.

A critical point for the audience to understand is that the helpers all run at the same speed, even when they are carrying water.

Suppose we run slower once we are carrying water. How does this extra facet influence the solution?  Note that you can solve this problem as well, however, but it is far more involved and involves calculus.