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?

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