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'Troublesome Triangles' printed from http://nrich.maths.org/

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Mudslides, avalanches and earthquakes have much in common - they involve physical environment which had appeared quite stable and then suddenly become very unstable, often with devastating consequences.  

In this game, students play a simple simulation game to investigate what causes equilibrium and instability in systems like these.

The Game

You need one person to be the Signaller, and two people to be Observers.  The rest of the class are the Movers.

The Basic Rules

  • The Movers all choose two other Movers WITHOUT telling anyone else who they have chosen.
  • The Movers then move around the room independently for a short while, stopping and standing quite still when told to stop by the Signaller.
  • The Signaller tells the Movers to move one pace to try to form an equilateral triangle with the two people they have chosen.
  • This continues, with Movers moving exactly one pace each time the signal is given.
  • The Observers count the number of signals/paces until everyone has formed their triangles - ie. equilibrium has occurred.

Try this several times, varying the roles.  

Questions to consider

  • What's the best way to record your findings?  Can you think of a good diagram?
  • How long does it take for equilibrium to occur?  
  • What is the shortest number of moves?  
  • What is the longest?  
  • What is the average?
  • What factors make a difference between a game when many moves are needed and one when only a few moves are needed?

Developing the game

There are several ideas here, and you may have ideas of your own.  Whatever you do to develop the game, do it systematically, recording carefully what you do and what the results are.

  • Once you have reached equilibrium, have one person away from their triangle - this person is a Disrupter.  How long does it take for equilibrium to be re-established?  What about if you have more than one Disrupter?
  • Pick some people to move only once, who then remain stationary for the rest of that game.  These people are Pins who are pinned down, impeding the movement of the rest.  How many Pins do you need for there to be a substantial change in the time it takes for equilibrium to occur?
  • What happens if you have some Disrupters and some Pins?
  • Divide the class into halves as nearly as possible.  Each Mover now has to choose people from the same group as themselves.  Before starting to move, mix everyone up.  Does this make a difference?
  • Now divide the class into two groups of varying sizes - how does this affect things?
  • Only let one group move at a time - so first one group moves, then the other.  
  • Divide people into Red and Green teams.  How does it change things if Movers pick one Red and one Green?
  • What about if Reds move quickly to their next positions, while Greens move very slowly?