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Why do this problem ?

This extended investigation problem will encourage creative hypothesis making and testing. Hypotheses can be tested out on the pattern maker giving instant feedback on whether a hypothesis needs to be rejected.The code system will encourage translation of vague visual statements into precise mathematical statements.

Possible approach

Get a feel for the interactivity and the resulting codes and grids. What patterns can we see? What differences or similarities? Stress that, whilst the computer uses a deterministic algorithm to determine the chance of a pattern being selected randomly this cannot ever be sure that a pattern was random or not in the way described. However, students can create firm hypotheses to test out the computer's algorithm. This can then be tested against the computer. Whilst it is not possible to accept a hypothesis with certainty, it is possible to reject a hypothesis with certainty if a run of the computer provides contradictory evidence.

Students will need to grapple with these ideas and should be prepared to mull over the problem over a period of time. Perhaps the patterns could be printed out and displayed for students to consider at their leisure.

Key questions

  • In what way are the patterns random?
  • What features do codes share and how do they differ?
  • How does changing a pattern slightly affect the code?
  • How might you group different patterns together? Would certain groupings of patterns be considered more or less likely than others?
  • Can you think of any patterns which would be very, very unlikely to be generated at radom?

Possible extension

Students might consider different ways in which they might choose to decide on the likelihood of a given pattern. Could they produce an algorithm? An example might be to declare a pattern unlikely if it contains less then 5 connected regions.

Possible support

Try the simpler problem Random Squares first.