The Princess's Problem

This is a problem from Greek mythology that has an intuitive solution but is rather more difficult to prove. The proof is developed for regular figures, and overall this is a problem in optimisation that uses formulae and inequalities, and could be used as an extension or reinforcement for the use of inequalities in linear programming. It is also a good discussion opener for problems in optimisation with constraints and with contexts in nature (e.g. what is the 3D shape that has minimum surface area for a fixed volume to enclose?).

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Summary

This is a problem from Greek mythology that has an intuitive solution but is rather more difficult to prove. The proof is developed for regular figures, and overall this is a problem in optimisation that uses formulae and inequalities, and could be used as an extension or reinforcement for the use of inequalities in linear programming. It is also a good discussion opener for problems in optimisation with constraints and with contexts in nature (e.g. what is the 3D shape that has minimum surface area for a fixed volume to enclose?).

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Mathematical strand

Algebra

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Prior knowledge

Appreciation of basic formulae for perimeter and area and an understanding of inequalities.

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Relevance to Core Maths qualifications

  • Eduqas
  • Pearson/ Edexcel

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Suggested approaches

Suitability for group or paired work with plenty of opportunities for collaboration and discussion, particularly related to problems in optimisation.

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Resources/documentation

In addition to this overview, there are:

  • Teacher notes
  • Handout with relevant background and questions posed

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Relevant digital technologies

This could usefully employ a spreadsheet for the IQN formulae for a regular plane shape with n sides and with increasing values of n.

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Possible extensions

There is plenty of scope for extending the problems introduced here, both in terms of the 2D shapes considered and analysed, extension to 3D shapes, and in changing the original problem from a ‘maximising area for given perimeter length’ to the dual problem of ‘minimising perimeter for a given area’.

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Acknowledgement

This is a resource developed for CMSP by David Burghes (CIMT, Plymouth University).